K 


M 


EMERSON'S  SECOND  PART. 


THE 


NORTH    AMERICAN 


ARITHMETIC. 


PART    SECOND, 


UNITlf 


ORAL  AND  WRITTEN  EXERCISES. 


HUNDRED 


TENS 


UNITS 

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138 


B  O  S  T  O  N% 
JENKS,    PALMER    ^    CO. 

SOLD     ALSO     BY 
ROE  LOCKWOOD,  NEW  YORK  ;    HOG  AN  Ic  1H(  ULADEL- 

PHIA  ;    GUSHING    .\:   BROTHER,   BALTIMOK  .  ,  ,,.   ,.  ,  MEECH, 
ST.  LOUIS  ;   WM.  C.  TABER  &  SON,  NEW  BEDFORD. 


55 


m 


EMERSON'S   SECOND    PART. 
THE 

NORTH    AMERICAN 

ARITHMETIC. 

PART    SECOND, 

UNITING 

ORAL  AND  WRITTEN  EXERCISES, 

IN 

CORRESPONDING   CHAPTERS. 


BY    FREDERICK    EMERSON, 

LITE  PRINCIPAL  IN  THE  DEPARTMENT  OP  ARITHMET«0, 
BOYLSTON  SCHOOL,  BOSTON. 


BOSTON: 

JENKS,    PALMER    &   CO. 

1848. 


Entered,  according  to  Act  of  Congress,  in  the  year  1832,  by  Frederick  Emerson^ 
Jn  the  Clerk's  Office  of  the  District  Court  of  the  District  of  Massachusetts. 


Orders  of  the  School  Committee  of  Boston, 

At  a  Meeting  of  the  School  Committee,  Nov.  18,  1834. 

Ordered,  That  Emerson's  North  American  Arithmetic,  Second 
and  Third  Parts,  be  substituted  in  the  Writing  Schools,  for  Gol- 
burn's  First  Lessons  and  Sequel.* 

Ordered,  That  the  Arithmetics  now  in  use  be  permittea  to  their 
present  owners ;  but  that  whenever  a  scholar  shall  have  occasion 
to  purchase  a  new  one,  the  North  American  Arithmetic  shall  be 
required.  ^^^^^^^    g  p  M'CLEARY,  Secretary. 

*  The  First  Part  was  already  adopted  by  a  previous  order. 


A  KEY  to  this  work,  containing  solutions  and  answers, 
[a  small  book  for  Teachers  only,]  is  published  separately 


PREFACE. 


This  book  is  intended  for  the  use  of  scholars  who  have 
been  taught  in  *  Part  First, '  or  by  some  other  means 
have  learned  to  add,  subtract,  and  multiply  numbers  as 
high  as  10,  mentally. 

The  whole  Course  of  Exercises,  of  which  this  is  the  Sec 
ond  Part,  has  been  divided  into  three  parts,  more  for  the 
sake  of  economy  and  convenience,  than  on  account  of  any 
natural  division  of  the  subject.  The  work  is  not  intended 
to  be  ai  record  of  the  science, — such  as  might  befit  the  pages 
of  an  encyclopedia, — but,  a  system  of  induction,  through 
which  the  scholar  may  be  led  to  the  discovery  of  arithmet- 
ical truth,  and  the  proper  application  of  arithmetical  ope- 
rations. Rules,  and  the  technical  language  necessary  to 
their  composition,  are  avoided  in  the  early  part  of  the 
course — they  are  not  introduced  until  the  learner  is  sup- 
posed prepared,  by  intellectual  improvement  from  previous 
lessons,  to  meet  them  understandingly. 

In  the  arrangement  of  the  exercises  in  this  volume,  I 
have  been  governed  by  the  natural  order  of  the  science; 
believing,  that  any  deviation  from  that  order,  with  a  view 
of  rendering  the  work  more  immediately  practical,  would 
render  it  in  reality  less  practical,  as  it  would  necessarily  lead 
the  scholar  into  a  habit  of  performing  operations,  without 
comprehending  the  principles  which  justify  them.  The 
first  six  chapters  consist  of  oral  exercises,  and  the  last  six 
of  correspondent  written  exercises.  The  work  may  there- 
fore be  viewed  as  two  entire  systems  of  arithmetic — Oral 
and  Written. 

Although  Part  Second  does  not  complete  the  series  of 
books,  entitled  *  The  North  American  Arithmetic,'  still  it 
contains  the  essential  principles,  and  the  common  applica- 
tion of  the  science.  Scholars,  therefore,  who  shall  be 
properly  conducted  through  this  volume,  will  have  acquir- 
ed a  knowledge  of  Arithmetic,  adequate  to  all  the  pur- 
poses of  common  business.  Part  Third  is  designed  for 
those,  whose  continuance  at  school  shall  afford  oppor- 
tunity for  prosecuting  a  more  extended  course  of  study 


4  PREFACE. 

The  mode  of  teaching  arithmetic,  and  the  text-books, 
used  for  the  purpose,  in  a  great  portion  of  our  country, 
are  radically  defective.  Much  of  arithmetic  is  practised 
at  school,  but  little  is  learned.  The  scholar  is  put  to 
ciphering  without  adequate  mental  preparation,  and  is  re- 
ferred to  the  direction  of  rules,  whose  phraseology  and 
principles  are  to  a  learner  equally  obscure.  By  a  tedious 
course  of  practice,  perhaps  he  acquires  a  certain  mechan- 
ical dexterity  in  performing  operations ;  but  no  sooner  does 
he  enter  upon  the  business  of  life,  than  he  abandons  the 
ru/es  of  his  book,  and,  in  his  own  way,  learns  so  much  of 
arithmetic  as  his  occupation  requires. 

Whether  the  following  treatise  is  calculated  to  afford 
any  remedy  for  the  defects  I  have  alluded  to,  others  will 
decide.  I  shall  spare  myself  the  task  of  a  prefatory  detail 
of  what  ^^the  author  conceives^'  to  be  its  advantages,  and 
will  only  add,  that  the  design  and  execution  of  the  work, 
have  cost  me  much  time  and  labor 

F.  Emerson. 

Boston,  January,  1832. 


NOTE  TO  TEACHERS. 

It  will  be  most  advantageous  for  young  scholars,  to  go  through  with  all  the 
Oral  Arithmetic  before  they  enter  upon  the  Written  Arithmetic.  Older  scholars. 
However,  after  performing  the  exercises  in  the  first  chapter  of  Oral  Arithmetic, 
may  pass  immediately  to  the  exercises  in  the  first  chapterof  Written  Arithme- 
tic; and  after  concluding  this  chapter,  may  take  up  the  two  second  chapters 
in  the  same  order ;  and  thus  proceed  through  the  book. 

Much  time  has  been  wasted  in  some  of  our  schools,  by  the  practice  of  teach- 
ing individually  f  instead  of  teaching  in  classes.  If  this  practice  has  been 
owing  in  any  degree  to  the  arrangement  of  text-books,  it  is  hoped  the  present 
arrangement  will  afford  a  remedy.  There  can  be  no  more  objection  to  a  distinct 
classification  of  a  school  for  the  purpose  of  teaching  arithmetic,  than  there  is  to 
a  like  classification  for  the  purpose  of  teaching  orthography:  and  the  advanta- 
ges of  class- instruction  in  the  former  branch,  are  as  great  as  those  in  the  latter. 

The  examples  contained  in  the  first  six  chapters,  do  not  require  tlieuseof  the 
elate.  The  answers,  with  the  process  of  obtaining  them,  and  the  reasons  whieb 
justify  the  process,  are  to  be  given  orally.  For  example,  the  following  question 
mav  be  supposed  to  give  rise  to  the  subjoined  exercise. 

jExample.  A  trader  purchased  9  barrels  of  flour,  at  7  dollars  a  barrel,  and 
Bold  the  whole  for  68  dollars.  What  did  he  gain  in  the  trade'?  Pupil.  *  He 
gained  five  dollars.'  Teacher.  *  How  do  you  perceive  itl'  Pupil.  '  If  one 
barrel  cost  seven  dollars,  nine  barrels  must  have  cost  nine  times  seven  dollars, 
which  is  sixty-three  dollars.  He  must  have  gained  the  difference  between 
iixty-three  dollars  and  sixty-eight  dollars.     63  from  68  leaves  5.' 

Learners  should  not  be  confined  to  any  form  of  expression  in  solutions— 
their  reasoning  should  be  their  own.  By  a  little  practice,  they  will  acquire 
an  aetoni^ihing  acuteness  of  apprehension,  and  facilitv  of  expression. 


ORAL    ARITHMETIC. 

CHAPTER   I. 
NUMERATIOIV. 

#  Section  1. 

When  we  have  a  large  number  of  articles  to  count,  such 
as  quills,  nuts,  cents,  &c.,  we  may,  if  we  please,  count 
them  by  tens.  Let  us  suppose  we  have  a  quantity  of 
cents  before  us,  and  proceed  to  count  them  as  follows. 

We  first  count  out  ten  cents,  and  lay  them  in  a  pile. 
We  then  count  out  ten  more,  and  lay  them  in  another  pile; 
then  ten  more  for  another  pile;  and  thus  we  continue  to 
count  out  ten  at  a  time,  until  we  have  counted  ten  piles. 
We  put  these  ten  piles  together,  and  they  make  a  large 
pile  containing  One  Hundred  cents. 

Again  we  count  out  ten  cents  at  a  time,  until  we  have 
counted  ten  small  piles,  as  before.  We  put  these  togeth- 
er, and  they  make  a  large  pile  containing  one  hundred, 
like  the  hundred  we  first  counted.  We  have  now  count- 
ed two  hundred  cents,  and  they  lie  in  two  large  piles. 

Having  learned  what  is  meant  by  two  hundreds^  we  pro- 
ceed to  count  out  one  hundred  cents  more;  and  after  pla- 
cing them  by  the  side  of  the  two  hundreds,  the  three  piles 
make  three  hundreds.  Four  large  piles  will  be  four  hun- 
dreds; five  piles  will  be  five  hundreds;  six  piles  will  be  six 
hundreds;  seven  piles  will  be  seven  hundreds;  eight  piles 
will  be  eight  hundreds;  nine  piles  will  be  nine  hundreds; 
and  when  we  have  counted  out  ten  of  these  piles,  we  put 
the  whole  together.  They  make  a  pile  still  larger,  and 
the  number  of  cents  contained  in  it  is  One  Thousand. 


6 


ORAL   ARITHiMiTlC. 


CHAP  I. 


Examine  the   arrangement   of  dots  enclosed    in  ihe 
lines  below,  and  find  how  many  there  are  in  each  en- 
closure.    Observe,  that  the  figures    standing   over  the 
several  enclosures,  represent  the  number  of  dots  con 
tained  therein. 


10 


100 


1000 


Example  1 .  Which  of  these  numbers  is  the  greatest. 
One,  or  Ten^  or  One  Hundred^  or  One  Thousand? 
2.  How  many  ones  are  there  in  a  ten  ? 


Sec.  2 


NUMERATION. 


3.  How  many  tens  are  there  in  a  hundred  ? 

4.  How  many  hundreds  are  there  in  a  thousand  ? 

5.  Ten  ones  make  what  number  ?  Ten  tens  make 
what  number  ?     Ten  hundreds  make  what  number  ? 

6.  What  figures  stand  to  represent  the  number  ten  ? 

7.  What  figures  stand  to  represent  one  hundred  ? 

8.  What  figures  stand  to  represent  one  thousand? 

Section  2. 

If  one  hundred  scholars  were  in  school,  and  one 
scholar  more  should  come  in,  the  number  of  scholars 
would  then  be  one  hundred  and  one;  and  would  be  ex- 
pressed in  figures  thus; — 101.  Again,  if  you  had  one 
hundred  books,  and  you  should  buy  two  books  more, 
you  would  then  have  one  hundred  and  two  books,  and 
their  number  would  be  expressed  in  figures  thus; — 102. 

In  Part  First,  you  learned  to  read  figures  expressing  all 
numbers,  from  One  to  One  Hundred,  You  will  now  see, 
m  the  following  columns,  how^  the  figures  stand  to  express 
numbers,  from  One  hundred^  to  Two  hundred. 


100  One  hundred, 

101  one  hund.  and 

102  one  hund.  and 

103  one  hund.  and 

104  one  hund.  and 

105  one  hund.  and 

106  one  hund.  and 

107  one  hund.  and 

108  one  hund.  and 

1 09  one  hund.  and 

110  one  hund.  and 

111  one  hund.  and 

112  one  hund.  and 

113  one  hund.  and 

114  one  hund.  and 

115  one  hund.  and 

116  one  hund.  and 

117  one  hund.  and 

118  one  hund.  and 

119  one  hund.  and 


one, 

two, 

three, 

four, 

five, 

six, 

seven, 

eight, 

nine, 

ten, 

eleven, 

twelve, 

thirteen, 

fourteen, 

fifteen, 

sixteen, 

seventeen, 

eighteen, 

nineteen, 


120  one  hund.  and  twenty, 

121  one  hund.  and  twenty-one, 

122  one  hund.  and  twenty- two, 

123  one  hund.  and  twenty-three 

1 30  One  hund.  and  thirty. 
140  One  hund,  and  forty. 
150  One  hund.  and  fifty. 
160  One  hund.  and  sixty. 
170  One  hund.  and  seventy. 
180  One  hund.  and  eighty. 
190  One  hund.  and  ninety. 
200  Two  hundred. 


8  ORAL    ARITHMETIC.  1 

Edward's  mother  gave  him  one  hundred  walnuts,  his 
sister  gave  him  sixty,  and  his  brother  gave  him  eight; 
making  together,  one  hundred  and  sixty-eight.  Being 
required  to  tell  what  figures  would  express  the  number  of 
his  walnuts,  Edward  looked  over  the  columns  of  figures 
on  the  last  page,  and  discovered,  (as  you  may),  that  1 
means  one  hundred^  whenever  two  figures  are  standing 
at  the  right  hand  of  it;  and,  that  6  means  sixty ^  whenever 
one  figure  is  standing  at  the  right  hand  of  it.  He  there- 
fore said,    "1,    6,    8,    are  the  figures." 

1.  How  many  tens  does  the  figure  6  represent,  when 
there  is  one  figure  standing  at  the  right  of  it  ? 

2.  What  are  6  tens  usually  called,  in  reading  numbers  } 

3.  How  many  tens  does  the  figure  4  represent,  when 
there  is  one  other  figure  standing  at  the  right  of  it  ? 

4.  What  are  4  tens  usually  called,  in  reading  numbers  ^ 

5.  What  number  does  the  figure  1  represent,  when 
there  is  one  other  figure  standing  at  the  right  of  it  ^ 

6.  What  number  does  the  figure  1  represent,  when 
there  are  two  other  figures  standing  at  the  right  of  it.'^ 

7.  What  are  1  hundred  and  5  tens  usually  called  ? 

8.  What  are  1  hundred  and  9  tens  usually  called.^ 

9.  What  are  1  hundred  and  3  ones  usually  called  ? 

10.  What  are  1  hundred  and  8  ones  usually  called  ? 

11.  What  are  8  tens  and  2  ones  usually  called  ^ 

12.  What  are  1  hundred,  and  7  tens,  and  5  ones  usu- 
ally called,  in  reading  numbers  ? 

Note  to  Teachers,     Require  the  learners  to  read  the  numbers  expressed  id 
the  following  columns,  without  recourse  to  the  preceding  columns. 


109 

172 

104 

168 

113 

127 

190 

110 

140 

147 

145 

121 

132 

122 

169 

163 

143 

155 

195 

183 

181 

165 

176 

177 

103 

118 

187 

198 

159 

125 

136 

154 

186 

131 

158 

The  comparisons  on  the  next  page  will  show  you^ 
that  all  the  hundreds  are  expressed  in  the  same  manner 
that  one  hundred  is  expressed. 


X.  ADDITION.  9 

100  One  hundreu.  133  One  huiid.  &  thirty-three. 

200  Two  hundred.  633  Six  hund.  &  thirty-three. 

106  One  hund.  &  six.  149  One  hund.  &  forty-nine. 

306  Three  hund.  &  six.  749  Seven  hund.  &  forty-nine. 

117  One  hund.  &  seventeen.      154  One  hund.  &  fifty-four. 
417  Four  hund.  &  seventeen.     854  Eight  hund.  &  fifty-four. 

121  One  hund.  &  twenty-one.     199  One  hund.  &  ninety-nine. 
621  Five  hund.  &  twenty-one.    999  Nine  hund.  &  ninety-nine. 

Note  to  Teachers.    The  learners  may  be  required  to  read  tlie  several 
numbers  expressed  in  the  following  columns  of  figures. 


814 

293 

552 

466 

851 

372 

947 

444 

664 

767 

528 

381 

786 

391 

579 

451 

619 

369 

940 

296 

CHAP.  11. 

ADDITIOIV. 

Section    1. 

1.  The  Humane  Society  gave  Charles  a  premium  of 
6  dollars,  for  saving  a  boy  from  drowning,  and  a  lady 
gave  him  5  dollars  more.     How  much  did  he  receive  } 

Solution,     6  dollars  and  5  dollars  are  11  dollars. 

2.  A  merchant  sold  7  barrels  of  flour  to  one  man,  and 
5  to  another.     How  many  barrels  did  he  sell  ? 

3.  If  you  should  pay  9  cents  for  a  book,  and  4  cents 
for  a  pencil,  how  much  would  you  pay  for  both  ? 

4.  A  farmer  paid  10  dollars  for  a  plough,  and  9  dol- 
lars for  a  harrow.     How  much  did  he  pay  for  both  ? 

5.  A  baker  bought  8  barrels  of  flour  of  a  merchant, 
and  8  more  of  a  miller.     How  many  did  he  buy  } 

6.  Thomas  gave  9  cents  for  a  purse,  and  had  7  cents 
left  to  put  in  it      How  many  cents  had  he  at  first  ? 

7.  A  farmer  sold  5  cows,  and  then  had  6  cows  left 
How  many  cows  had  he  at  first? 


10  ORAL    ARITHMETIC.  II. 

8.  If  you  should  receive  9  dollars  from  one  man,  and 
5  from  another,  how  many  dollars  would  you  receive  ? 

Section    2. 

1.  Two  little  boys  went  into  a  shop  to  be  weighed. 
The  oldest  of  them  weighed  40  pounds,  and  the  youngest, 
30  pounds.  How  many  pounds  would  they  weigh  both 
together  ? 

Solution,  40  is  the  same  as  4  tens^  and  30  is  the  same 
as  3  tens.  Then  4  tens  and  3  tens  are  7  tens; — and  7 
tens  are  the  same  as  70. 

2.  There  were  40  oranges  in  one  basket,  and  20  in 
another.     How  many  were  there  in  both  baskets.'* 

3.  What  is  the  whole  number  of  scholars  in  a  school, 
that  consists  of  20  boys  and  30  girls  ? 

4.  A  baker  paid  50  dollars  for  a  horse,  and  30  dollars 
for  a  cart.    -How  many  dollars  did  he  pay  for  both  ? 

5.  If  I  read  50  pages  of  history,  and  40  pages  of 
poetry,  how  many  pages  do  I  read  of  both  ? 

6.  If  a  man  has  lived  20  years  in  the  city,  and  10 
years  in  the  country,  how  old  must  he  be  ? 

7.  James  paid  60  cents  for  his  Reader,  and  40  for 
his  Arithmetic.     How  many  cents  did  they  both  cost  ? 

8.  Suppose  you  should  buy  60  quills  at  one  store, 
and  50  at  another;  how  many  quills  would  you  have  ? 

Solution,  60  is  6  tens,  and  50  is  5  tens.  6  tens  and 
5  tens  are  11  tens.  11  tens  are  1  hundred  and  1  ten; — 
that  is,  110. 

9.  Suppose  70  books  are  upon  my  table,  and  I  put 
on  50  more;  how  many  will  then  be  on  the  table  ? 

10.  If  a  gold  watch  cost  90  dollars,  and  the  chain  40 
dollars;  how  many  dollars  do  they  both  cost.'* 

11.  In  a  certain  orchard,  there  are  80  pear  trees  and 
60  peach  trees.     How  many  trees  in  the  orchard  ? 

12.  If  90  persons  should  enter  a  hall  at  one  door,  and 
60  at  another;  how  many  would  there  be  in  the  hall.'* 

13.  If  I  purchase  80  barrels  of  flour  from  one  man, 
and  80  from  another;  how  many  barrels  shall  I  have  ^ 

14.  A  miller  had  90  bags  of  wheat  on  hand,  and  re* 
ceived  80  bags  more.     How  many  bags  had  he  then  ? 


9.  3.  ADDITION.  11 

15.  If  a  horse  cost  90  dollars^  and  a  gig  90  dollars, 
how  much  do  the  horse  and  gig  both  cost  ? 

16.  How  many  pounds  of  honey  in  two  jars; — there 
being  70  pounds  in  one  jar,  and  60  in  the  other  ? 

Section  3. 

1.  A  gardener  called  three  boys  to  the  garden  gate  to 
give  them  some  grapes.  To  the  first  boy  he  gave  40 
grapes,  and  to  the  second  40;  but  the  third  boy  attempted 
to  push  the  others  aside,  and  the  gardener  seeing  it,  gave 
him  only  6.     How  many  did  he  give  them  all  ? 

Solution.  40  grapes  and  40  grapes  are  80  grapes. 
Then  80  grapes  and  6  grapes  are  86  grapes. 

2.  John,  James,  and  Henry  went  a  fishing.  John 
caught  30  fishes,  and  James  caught  40;  bu^t  Henry  caught 
only  9.     How  many  did  they  all  catch  ? 

3.  How  many  are  30  and  40  and  9  ? 

4.  A  traveller  gave  70  dollars  for  his  horse,  20  dollars 
for  his  saddle,  and  5  dollars  for  his  bridle.  How  many 
dollars  did  he  give  for  the  whole  ? 

5.  How  many  are  70  and  20  and  5  ? 

6.  A  farmer  kept  50  sheep  m  one  pasture,  30  in 
another,  and  7  in  another.  If  he  had  kept  them  all  in  one 
pasture,  how  many  would  there  have  been  together  ? 

7.  How  many  are  50  and  30  and  7  ? 

8.  How  many  cents  will  it  take  to  buy  a  seal,  a 
blank-book,  and  a  pencil;  supposing  the  seal  to  cost  60 
cents,  the  blank-book  20  cents,  and  the  pencil  8  cents  ? 

9.  How  many  are  60  and  20  and  8  ? 

10.  An  escort  went  out  to  meet  Gen.  Layfayette:  40 
men  rode  on  horseback,  30  rode  in  gigs,  and  10  rode  in 
coaches.     Of  how  many  did  the  escort  consist? 

11.  How  many  are  40  and  30  and  107 

12.  How  many  are  40  and  30  and  3  r 

13.  How  many  are  60  and  20  and  5  i 

14.  How  many  are  30  and  30  and  7  i 
*     15.  How  many  are  50  and  40  and  9  i 

16.  How  many  are  50  and  50  and  8  i 

17.  How  many  are  60  and  40  and  4.? 

18.  How  manv  are  70  and  30  and  6  f 


12  ORAL    ARITHMETIC.  XL 

Section  4. 

1.  A  certain  class  consists  of  11  studious  boys,  and  2 
idle  boys.     How  many  are  there  in  the  class  ? 

2.  Hovvmanyarelland2?    11  and 3?  11  and 4?  Hand 
5?  llandG?  Hand??  llandS?   Iland9?  llandlO? 

3    Alfred  paid  11  cents  for  a  pen-knife,  and  10  cents 
for  a  writing-book.      How  much  did  he  pay  for  both  ? 

4.  If  you  should,  pay  12  cents  for  a  slate,  and  3  cents 
for  an  orange,  how  many  cents  would  they  both  cost  ? 

5.  How  many  are  12  and  2?  12and3?  12and4?  12  and 
5.^   12and6?  12and7?  12and8?  12and9?  12andl0? 

6.  If  12  boys  play  at  foot-ball  on  one  side,  and  8  boys 
on  the  other,  how  many  are  there  in  the  play  ? 

7.  A  certain  class  consisted  of  13  small  boys,  and  4 
large  boys.     How  many  were  there  in  the  class  ? 

8.  How  many  are  13  and  2?  13and3?  13and4?  13and 
5?  13and6?  13and7?  13and8?  13and9?  13andl0? 

9.  A  number  of  sheep  are  in  a  fold; — 13  are  lying 
down,  and  6  are  standing  up.     How  many  are  there  ? 

10.  There  were  14  hats  hanging  up,  and  5  more  *ying 
down.     How  many  hats  were  there  in  all  ? 

11.  How  many  are  14and2?    14and3?    14and4?    14and 
5?    14and6?    14and7?    14and8?    14and9?    14andl0r 

12.  If  you  give  14  cents  for  a  bow,  and  4  cents  for 
an  arrow,  how  much  do  the  bow  and  arrow  cost? 

13.  A  wagoner  drove  15  miles  in  the  forenoon,  and  6 
in  the  afternoon.     How  many  miles  in  the  day  ? 

14.  How  many  are  15and2.'^    15and3?    15and4?    15and 
5?    15and6?    15and7.^    15and8.^    15and9?    15andlO.? 

15.  If  a  cow  be  worth  15  dollars,  and  a  sheep  2  dol- 
lars, what  are  the  cow  and  sheep  together  worth  ? 

16.  David  wrote  16  fines  in  the  forenoon,  and  7  in  the 
afternoon.     How  many  lines  did  he  write  in  the  day.'* 

17.  How  many  are  16and2?    16and3?    16and4?    16and 
6.?    16and6?    16and7?    16and8?    16and9.?    16andl0.? 

18.  A  trooper  gave  16  dollars  for  his  saddle,  and  9 
dollars  for  his  bridle.     How  much  did  he  pay  for  both  ? 


4.  5.  ADDITION.  It 

19.  A  man  lost  S  doJars,  and  still  had  17  dollars  left. 
How  many  dollars  had  he  before  he  lost  any  ? 

20.  Howmanyarel7and2?    17and3?  17and4?  17and 
5?     17and6?     17and7?    llandS?    17and9?    17andl0.? 

21.  If  a  time-piece  cost  17  dollars  and  a  looking-glas«? 
7  dollars,  how  many  dollars  do  they  both  cost  ? 

22.  While  18  doves  were  upon  a  roof,  9  doves  more  lit 
among  them.     How  many  were  then  upon  the  roof? 

23.  Howmanyarel8and2?  18and3?  18and4?    18and 
5?  18and6?    18and7?    18and8?    18and9?    18and'10? 

24.  A  man  rolled  18  barrels  of  flour  out  of  a  mill  and 
a  boy  rolled  out  5  more.     How  many  did  both  roll  out  ? 

25.  A  young  man  began  studying  law  at  the  age  of  19 
vears,  and  studied  3  years.     At  what  age  did  he  finish  ? 

26.  Howmanvarel9and2?    19and3?    19and4?    19and 
5?    19and6?    19and7?    19and8?    19and9?    19andlO? 

27.  A  farmer  mixed  19  bushels  of  oats  with  10  of  corn. 
How'many  bushels  were  there  of  the  mixture  ? 

Note  to  Teachers.     Tlie  following   combinations  may  be  embraced  in 
Beparate  questions  by  the  teacher ;  thus, — How  many  are  19  and  4  ? 


19  and  4 

14  and  2 

19  and  9 

15  and  9 

16  and  3 

19  and  6 

16  and  8 

13  and  7 

18  and  5 

16  and  6 

13  and  5 

14  and  4 

12  and  2 

15  and  4 

18  and  8 

14  and  8 

17  and  7 

18  and  2 

14  and  6 

12  and  5 

19  and  7 

17  and  6 

12  and  9 

14  and  7 

15  and  5 

19  and  3 

13  and  3 

17  and  8 

19  and  8 

16  and  5 

12  and  7 

12  and  6 

17  and  9 

15  and  8 

16  and  6 

16  and  9 

12  and  8 

16  and  4 

17  and  5 

1 3  and  8 

Section  5. 

1.  Charles  had  25  books  in  his  library,  and  his  falhe 
gave  him  8  more.     How  many  had  he  then  ? 

Suggestion.  You  will  easily  perceive  how  many  .25. 
and  8  are,  since  you  already  know  that  5  and  8  are  13, 
and  that  15  and  8  are  23. 


14  ORAL    ARITHMETIC.  II. 

2.  A  father  said  to  his  son, '  You  are  7  years  old,  and  I 
am  47 — How  old  shall  we  each  of  us  become,  In  9  years 
from  this  time?'     What  should  have  been  the  answer? 

3.  James  bought  a  small  book  for  6  cents,  and  David 
bought  a  large  book  for  56  cents.  For  how  many  cents 
must  each  boy  sell  his  book,  in  order  to  get  4  cents  more 
than  he  gave  ? 

4.  Julia  was  returning  from  a  walk  in  the  garden,  with 
8  red  roses,  and  63  white  roses.  She  met  her  brother, 
who  gave  her  6  more  red  roses,  and  6  white  ones.  How 
many  of  each  kind  had  she  then? 

5.  William  has  9  cents,  and  John  has  79  cents.  If 
they  should  each  of  them  get  10  cents  more,  how  many 
would  each  boy  then  have  ? 

Note  to  Teachers.     The  following  combinations  maybe  embraced  inques- 
tions  by  the  teacher;   thus, — How  many  are  3  and  9  ? 


3  and  9 

54  and  6 

41  and  10 

38  and   8 

13  and  9 

67  and  8 

53  and    9 

41  and   7 

7  and*  6 

72  and  9 

65  and    6 

53  and    9 

27  and  6 

85  and  7 

77  and    3 

65  and   2 

8  and  8 

90  and  5 

89  and    5 

77  and  10 

38  and  8 

IS  and  4 

92  and    7 

89  and    6 

1  and  7 

26  and  7 

14  and  10 

92  and   4 

41  and  7 

39  and  2 

26  and   7 

14  and   8 

Section  6. 

1.  A  trader  paid  29  dollars  for  a  chest  of  tea,  4  dollars 
for  a  box  of  lemons,  and  5  dollars  for  a  box  of  raisins. 
What  did  he-  pay  for  the  whole  ? 

2.  How  many  are  29  and  4  and  5  ? 

3.  If  I  pay  38  dollars  to  one  man,  6  to  another,  and  3 
to  another,  how  many  dollars  do  I  pay  out  ? 

4.  How  many  are  38  and  6  and  3  ? 

5.  Stephen  had  47  books;  he  bought  5  more,  and  then 
his  uncle  gave  him  6  more.     How  many  had  he  at  last  ? 

6.  How  many  are  47  and  5  and  6? 

7.  On  a  certain  day,  a  passenger  travelled  56  miles  in 
the  stage,  4  miles  in  a  wagon,  and  7  miles  on  foot.  How 
many  miles  did  he  travel  on  that  day? 

8.  How  many  are  56  and  4  and  7-^ 


6  7.  ADDITION.  15 

9.  If  a  yoke  of  oxen  be  worth  65  dollars,  a  sheep  8  dol- 
lars, and  a  lamb  2  dollars,  how  much  are  they  all  worth? 

10.  How  many  are  65  and  8  and  2? 

11.  A  school  boy  paid  74  cents  for  a  reading  book, 

7  cents  for  a  writing  book,  and  9  cents  for  some  quills. 
How  many  cents  did  he  pay  for  the  whole  ^ 

12.  How  many  are  74  and  7  and  9  ? 

13.  A  meeting  was  held. in  a  country  village,  to  which 
83  persons  walked,  9  rode  on  horseback,  and  8  rode  in 
gigs.     How  many  attended  the  meeting  ? 

14.  How  many  are  83  and  9  and  8? 

15.  A  market  man  received  92  dollars  for  butter,  9 
dollars  for  cheese,  and  5  dollars  for  poultry.  How  many 
dollars  did  he  receive  for  the  whole  ? 

16.  How  many  are  92  and  9  and  5  } 

Section   7. 

1.  The  captain  of  a  steam-boat  received  the  following 

fassengers; — 45  gentlemen,  20  ladies,  and  8  children, 
low  many  passengers  were  there  in  all  ? 
Solution.     45  and  20  are  65;  then  65  and  8  are  73. 
Answer^  73  passengers. 

2.  If  a  quire  of  paper  cost  23  cents,  a  book  30  centS; 
and  a  pencil  9  cents,  what  do  they  all  cost } 

3.  How  many  are  23  and  30  and  9? 

4.  Alfred  paid  25  cents  for  his  penknife,  and  20  cents 
for  his  wallet,  and  then  had  5  cents  left.  How  many 
cents  had  he  at  first  ? 

5.  How  many  are  25  and  20  and  5  } 

6.  A  lady  gave  57  cents  for  a  fan,  30  cents  for  a  work 
bag,  and  4  cents  for  some  needles.  How  m^ny  cents 
did  she  lay  out  ? 

7.  How  many  are  57  and  30  and  4  ? 

8.  A  fowler  went  out  one  morning  to  shoot  birds; — 
he  shot  46  plovers,  50  snipes,  and  6  quails.  How  many 
birds  did  he  shoot  ? 

9.  How  many  are  46  and  50  and  6  ? 

10.  If  a  cart  cost  26  dollars,  a  plough  10  dollars,  and 
i  chain  5  dollars,  what  do  they  all  cost  ? 

11.  How  many  are  26  and  10  and  5  } 


16  ORAL    ARITHMETIC.  III. 

12.  A  farmer  sold  a  horse  for  75  dollars,  a  cow  for  30 
dollars,  and  a  sheep  for  5  dollars.  How  many  dollars  did 
he  get  for  the  whole  ? 

13.  How  many  are  75  and  3C  and  5  ? 

14.  Wilham  gave  64  cents  for  a  handkerchief  and  40 
cents  for  a  pair  of  gloves,  and  then  had  9  cents  left. 
How  many  cents  had  he  at  first  ? 

15.  How  many  are  64  and  40  and  9  ^ 

16.  How  many  are  5  and  9  and  2  and  8  and  6  and  4? 

17.  How  many  are  8  and  3  and  7  and  6  and  5  and  3.^ 

18.  How  many  are  6  and  8  and  4  and  9  and  7  and  5.'* 

19.  How  many  are  9  and  7  and  2  and  8  and  3  and  6.'' 

20.  How  many  are  17  and  5  and  0  and  9  and  6  and  8.'^ 

21.  How  many  are  23  and  8  and  1  and  0  and  9  and  7? 

22.  How  many  are  48  and  6  and  7  and  4  and  0  and  2? 

23.  How  many  are  71  and  3  and  9  and  0  and  6  and  9? 


CHAP.  III. 
SUBTRACTION* 

Section  1. 

1.  There  were  9  passengers  in  a  stage;  3  of  them 
got  out  to  walk:  how  many  remained  in  the  stage  ? 

Solution.     3  from  9  leaves  6.    Answer.  6  passengers. 

2.  A  boy  having  10  cents,  paid  6  cents  for  a  kite,  and 
*ost  the  remainder.     How  much  did  he  lose  ? 

3.  Ann  has  12  books  andJuHahas7.     Howmanymore 
must  Julia  have,  to  make  her  number  equal  to  Ann's  ? 

4.  Andrew  has  11  cents,  and  James  has  only  5  cents. 
How  many  cents  has  Andrew  more  than  James  ? 

5.  Stephen  has  8  cents,  and  wishes  to  buy  a  knife  worth 
16  cents.     How  many  more  cents  does  he  want? 

6.  A  lady  went  to  buy  goods,  carrying  13  dollars;  she 
returned  with  9  dollars.      How  much  did  she  spend  ? 

7.  A  merchant  bought  a  box  of  goods  for  10  dollars, 
and  sold  it  for  14  dollars.     How  much  did  he  gain  ? 


1.  2.  SUBTRACTIOIN.  17 

8  Jonathan  is  7  years  old,  and  his  brother  is  11  years 
old.     What  is  ^he  difference  in  their  ages  ? 

9.  Henry  bought  a  book  and  a  pencil  for  19  cents:  he 
gave  10  cents  for  the  book;  what  did  the  pencil  cost? 

10.  A  man  who  owed  a  debt  of  12  dollars,  paid  5  dol- 
lars of  it.     How  many  dollars  remained  unpaid.'* 

11.  John  sold  a  knife  for  18  cents,  which  was  9  cents 
more  than  he  gave  for  it.     How  much  did  he  give  for  it  ? 

12.  A  farmer  agreed  to  give  17  dollars  for  a  cow:  and 
he  paid  S  dollars  down.      How  much  did  he  still  owe  ? 

Section  2. 

1 .  A  sloop  of  war  went  out  with  a  crew  of  70  men,  and 
fell  into  an  engagement,  in  which  30  of  her  men  were 
killed.     How  many  of  the  crew  were  still  living  ? 

Suggestion.  Consider  the  numbers  to  be,  7  tens^  and 
3  tens; — you  may  then  take  30  from  70  as  easily  as  you 
can  take  3  from  7. 

2.  A  market  woman  had  60  oranges,  and  sold  20  of 
them.     How  many  had  she  remaining  ? 

3.  20  from  60  leaves  how  many  ?  How  many  are  20 anrf 40  ? 

4.  A  certain  school  consists  of  50  scholars,  30  of 
whom  are  girls.     How^  many  boys  are  there  ? 

5.  30  from  50 leaves  how  many  ?    Hoic  many  are  30  and  20  ? 

6.  A  baker  had  80  dollars  to  lay  out  for  a  horse  and 
cart.  After  having  paid  50  dollars  for  a  horse,  how  many 
dollars  had  he  left  to  purchase  the  cart  ? 

7 .  50  from  80  leaves  how  many  ?  Hoio  many  are  50  and  30  ? 

8.  If  your  lesson  for  the  whole  day  be  40  questions  in 
ihis  book,  and  you  answer  20  questions  in  the  forenoon, 
how  many  are  there  left  for  the  afternoon  } 

9.  20  from  40  leaves  how  many  ?   How  many  are  20  am/ 20  ? 

10.  I  have  read  40  pages,  in  a  book  which  contains  90 
pages.     How  many  pages  remain  to  be  read  ^ 

11.  40  from  90  leaves  how  many.'*  Hoiv  many  are  40  and  50? 

12.  James  had  70  cents,  and  paid  40  of  them  for  a 
school-book.     How  many  cents  ha'd  he  left  ^ 

1 3.  40  from  70  leav  es  how  ma'hy  ?  How  many  are  40  arw?  30? 


18  ORAL    ARITHMETIC.  HI 

Section  3. 

1.  A  man  received  12  dollars  for  work,  and  paid  2 
dollars  for  his  board.     How  many  dollars  did  he  save  ? 

2.  How  many  will  remain,  if  we  take  2  from  12  ?  2 
from  13?  2froml4?  2from  15  ?  2 from  16  ?  2  from 
17?     2  from  18?     2  from  19?     2  from  20  ? 

o.  A  stable  keeper  owned  14  fine  horses.  After  sell- 
ing off  3  of  them,  how  many  had  he  remaining  ? 

4.  How  many  will  remain,  if  we  take  3  from  13  ?  3 
from  14?    3  from  15?    3  from  16?    3  from  17?    3  from 

18  ?     3  from  19  ?     3  from  20  ?     3  from  21  ? 

5.  16  boys  were  dismissed,  but  4  of  them  were  called 
back  for  being  noisy.     How  many  were  allowed  to  go? 

6.  How  many  will  remain,  if  we  take  4  from  14  ?  4 
from  15?    4  from  16?    4  from  17?    4  from  18?    4  from 

19  ?     4  from  20  ?     4  from  21  ?     4  from  22  ? 

7.  A  man,  who  had  18  dollars,  paid  5  dollars  for  a 
pair  of  boots.     How  many  dollars  had  he  remaining? 

8.  How  many  will  remain,  if  we  take  5  from  15  ?  5 
from  16  ?     5  from  17?     5  from  18?     5  from  19?     5  from 

20  ?     5  from  21  ?     5  from  22  ?     5  from  23  ? 

9.  If  you  had  just  20  cents,  and  you  should  lose  6 
cents,  how  many  cents  would  you  then  have  ? 

10.  How  many  will  remain,  if  we  take  6  from  16?  6 
from  17?     6  from  18?    6  from  19?    6  from  20  ?    6  from 

21  ?     6  from  22  ?     6  from  23  ?     6  from  24  ? 

11.  A  man  who  had  22  dollars  on  hand,  lent  7  dollars 
to  his  neighbour.     How  many  dollars  had  he  remaining  ? 

12.  How  many  will  remain,  if  we  take  7  from  17?  7 
from  18  ?    7  from  19  ?    7  from  20  ?    7  from  21  ?    7  from 

22  ?     7  from  23  ?     7  from  24  ?     7  from  25  ? 

13.  24  peaches  grew  upon  a  young  peach  tree,  and  the 
owner  took  off  8  of  them.     How  many  remained  on? 

14.  How  many  will  remain  if  we  take  8  from  18?  8 
from  19?     8  from  20  ?     8  from  21?     8  from  22  ?     8  from 

23  ?     8  from  24  ?     8  from  25  ?     8  from  26  ? 


3.4.  CORRESPONDENT    EXAMPLES.  19 

15.  Suppose  you  had  26  cents,  and  paid  9  of  them 
for  a  dozen  of  quills;  how  manj  cents  have  you  left.^ 

16.  How  many  will  remain,  if  we  take  9  from  19  .'^  9 
from  20?  9  from  21?  9  from  22?  9  from  23?  9  from 
24  ?     9  from  25  ?     9  from  26  ?     9  from  27  ? 

17.  James  received  28  cents,  and  Charles  received  10 
cents  less  than  James.     How  many  did  Charles  receive  ? 

18.  Howmany  will  remain,  if  we  take  10 from  20?  10 
from21?  10from22?  10from23?  10from24?  lOfrom 
25?     10  from  26?     10  from  27?     10  from  28? 

Note  to  Teachers.     The  following  combinations  of  numbers  may  be  em- 
braced in  quesions  by  the  teacher,  thus, — 2  from  17  leaves  how  many? 

2  from  17        8  from  19        7  from  24        5  from  23 

7  from  21  3  from  20  9  from  21  6  from. 24 
9  from  24        9  from  26         3  from  18         8  from  23 

3  from  16        7  from  18        8  from  22        3  from  21 

8  from  26  2  from  15  6  from  17  9  from  27 
6  from  19  6  from  22  5  from  18  7  from  25 
5  from  22  5  from  19  4  from  16  2  from  20 

4  from  20  4  from  15  2  from  19  4  from  22 
10  from  25  10  from  27  10  from  26  10  from  29 

Section   4. 

CORRESPONDENT    EXAMPLES. 

1.  A  farmer,  who  had  19  dollars  on  hand,  received  5 
dollars  for  a  sheep.     How  many  dollars  had  he  then  ? 

2.  A  butcher,  who  had  24  dollars  on  hand,  paid  out  5 
dollars  for  a  sheep.     How  many  dollars  had  he  left? 

3.  A  jeweller  gave  17  dollars  for  a  silver  watch,  and 
sold  it  for  6  dollars  more  than  he  gave  for  it.  For  how 
many  dollars  did  he  sell  it  ? 

4.  A  young  man  gave  23  dollars  for  a  watch,  and  was 
obHged  to  sell  it  for  6  dollars  less  than  he  gave.  For 
how  much  did  he  sell  it  ? 

5.  It  was  4  years  ago,  that  Samuel  left  the  academy, 
and  he  was  then  15  years  o.d.     How  old  is  he  now  ? 

6.  Sarah  is  19  years  old;  her  father  died  when  she 
was  15.     How  manv  years  is  it,  since  her  father  died  ? 


20  ORAL    ARITHMETIC.  III. 

7.  A  farmer  who  had  26  sheep,  purchased  8  more. 
How  many  sheep  had  he  then  ? 

8.  A  farmer  who  had  34  sheep,  sold  8  of  his  flock. 
How  many  had  he  remaining  ? 

9.  A  merchant  who  had  9  dollars  on  hand,  received 
27  dollars  more,  for  a  quantity  of  goods.  How  many  dol- 
lars had  he  then  ? 

10.  A  merchant  who  had  36  dollars  in  his  pocket,  paid 
a  small  debt,  and  then  had  27  dollars  left.  How  many 
dollars  did  he  pay  ? 

11.  A  wagon  passed  along,  carrying  38  empty  barrels 
and  7  full  ones.     How  many  barrels  in  all } 

12.  In  a  store-room  there  were  45  barrels,  only  7  of 
which  w^ere  filled.     How  many  were  empty .^ 

13.  Edward  paid  46  cents  for  a  book,  and  then  had  9 
cents  left.  How  many  cents  had  Edward  before  he 
purchased  the  book  ^ 

14.  Joseph's  father  gave  him  55  cents,  to  buy  a  book, 
but  he  obtained  the  book  for  46  cents.  How  many  cents 
did  Joseph  save  ? 

15.  How  many  are  57  and  5.*^  Then  if  we  take  5 
from  62,  how  many  remain } 

16.  How  many  are  64  and  6.^  Then  if  we  take  6 
from  70,  how  many  remain  ^ 

17.  How  many  are  79  and  8.^  Then  if  we  take  8 
from  87,  how  many  remain  } 

18.  How  many  are  86  and  6  }  Then  if  we  take  6 
from  92,  how  many  remain  } 

19.  How  many  are  48  and  9.'^  Then  if  we  take  9 
from  57,  how  many  remain  ^ 

20.  How  many  are  75  and  7  ^  Then  if  we  take  7 
from  82,  how  many  remain  ^ 

21.  How  many  are  36  and  5  } — how  many  are  5  and 
36  i  Then  5  from  41  leaves  how  many  } — 36  from  41 
leaves  how  m.any? 

22.  How  many  are  43  and  9  } —  how  man}^  are  9  and 
43  ?  Then  9  from  52  leaves  how  many? —  43  from  52 
.feaves  how  many } 


5.  MISCELLANEOUS    EXAMPLES.  21 

23.  How  many  are  54  and  6  ? —  how  many  are  6  and 
54  ?  Then  6  from  60  leaves  how  many  ? —  54  from  60 
leaves  how  many? 

24.  How  many  are  68  and  4? — how  many  are  4  and 
68  ^  Then  4  from  72  leaves  how  many? — 68  from  72 
leaves  how  many? 

25.  How  many  are  79  and  8? — how  many  are  8  and 
79?  Then  8  from  87  leaves  how  many? — 79  from  87 
leaves  how  many? 

26.  How  many  are  87  and  5? — how  many  are  5  and 
87?  Then  5  from  92  leaves  how  many.'* — 87  from  92 
leaves  how  many? 

Section   5. 

MISCELLANEOUS    EXAMPLES. 

1.  George  and  David  went  out  to  gather  lilies;  George 

fot  56,  and  David  49.     On  the  way  home,  George  gave 
)avid  8.     How  many  had  each  boy  then  ? 
Solution,     At  first,  George  had  56;  he  gave  away  8; 
8  from  56  leaves  48 At  first,  David  had  49;  he  re- 
ceived 8  more;  49  and  8  are  57. 

2.  A  clerk  went  out  to  collect  some  money.  He  re- 
ceived 60  dollars  from  one  man,  9  dollars  from  another, 
and  20  from  another ;  and  he  paid  a  debt  of  7  dollars. 
How  many  dollars  had  he  to  bring  in  ? 

Solution.  60  and  9  are  69,  and  20  are  89; — this  is 
the  number  of  dollars  he  collected.  He  then  paid  7 
dollars.     7  from  89  leaves  82. 

3.  Harriet  answered  23  questions  in  arithmetic,  and 
Mary  answered  7  more  than  Harriet.  How  many  ques- 
tions did  they  both  answer  ? 

4.  Edward  answered  36  questions  in  arithmetic,  and 
Stephen  answered  6  less  tb^a  Edward.  How  many 
questions  did  they  both  ansv.  er  ? 

5.  A  blacksmith  who  had  100  dollars,  laid  his  money 
out  as  follows — For  iron  6Q  dollars,  for  steel  30  dollars, 
and  the  remainder  for  coal.  How  many  dollars  did  he 
pay  for  coal  '^ 


22  ORAL    ARITHMETIC.  Ill 

6.  A  carpenter  paid  31  dollars  for  boards,  10  dollars 
for  shingles,  6  dollars  for  nails,  and  5  dollars  for  screws. 
How  many  dollars  did  he  spend  ? 

7.  A  trader  gave  4S  dollars  for  a  chest  of  tea,  and  3 
dollars  for  getting  it  home.  For  how  much  must  he  sell 
the  tea,  in  order  to  gain  8  dollars  ? 

8.  If  I  have  70  dollars  on  hand,  and  pay  out  4  dollars 
to  one  man,  20  to  another,  and  30  to  another,  how  many 
dollars  shall  I  have  remaining  ? 

9.  A  gentleman  travelled  8  miles  before  breakfast,  30 
more  before  dinner,  and  40  more  after  dinner.  How 
many  miles  did  he  travel  during  the  day  ? 

10.  A  merchant,  who  had  80  barrels  of  flour,  sold  to 
one  man  52  barrels,  to  another  6  barrels,  and  to  another 
5  barrels.     How  many  barrels  had  he  left  .^ 

11.  Ohver's  penknife  is  worth  33  cents,  and  Edwin's 
is  worth  only  19  cents.  Now  if  they  exchange  pen- 
knives, how  many  cents  must  Edwin  give  Oliver  ? 

12.  Arthur's  penknife  was  worth  25  cents,  and  Wal- 
ter's was  worth  only  18  cents:  still,  A.  gave  W.  6  cents 
to  exchange.     How  much  did  A.  lose  ? 

13.  On  the  Fourth  of  July,  Robert  had  50  cents  given 
him.  He  spent  6  cents  for  fruit,  12  cents  for  confection- 
ary, 20  cents  for  a  picture  of  Gen.  Washington,  and  gave 
away  5  cents.     How  many  cents  had  he  left? 

14.  Leonard  has  32  cents,  and  Albert  has  49.  How 
many  cents  has  Albert  more  than  Leonard  ? 

15.  Francis  being  asked  how  old  he  was,  answered, 
that  in  14  years  more,  he  should  be  25  years  old.  How 
old  was  he,  at  the  time  he  was  asked  ? 

16.  If  a  cow  be  worth  22  dollars,  and  a  calf  5  dollars, 
how  much  more  is  the  cow  worth  than  the  calf? 

17.  A  jockey  gave  85  dollars  for  a  horse,  and  sold  him 
for  68  dollars.     How  much  did  he  lose  ? 

18.  A  trader  gave  83  dollars  for  a  hogshead  of  sugar, 
and  sold  it  for  96  dollars.     How  much  did  he  gain  ? 

3  9.  A  farmer  gave  24  dollars  for  a  cart,  and  12  dollars 
for  a, plough.     How  many  dollars  did  both  cost? 

20.  If  a  gold  watch  be  worth  64  dollars,  and  a  gold 
chain  18  dollars,  how  much  are  they  both  worth? 


5.  MISCELLANEOUS    EXAMPLES.  23 

21.  A  man  gave  13  dollars  for  the  improvenienl  of  a 
piece  of  ground,  paid  36  dollars  for  having  it  cultivated, 
and  then  sold  the  produce  for  48  dollars.  How  many 
dollars  did  he  lose  ? 

22.  A  market-man  bought  some  butter  for  S  dollars, 
some  cheese  for  15  dollars,  and  some  poultry  for  12 
dollars;  and  then  sold  the  whole  for  39  dollars.  Did  he 
gain  or  lose  ? —  and  how  much  ? 

23.  John  bought  a  penknife  for  25  cents;  he  exchanged 
It  for  a  better  one,  paying  16  cents,  and  then  sold  the 
better  one  for  40  cents.  Did  he  gain  or  lose  ? —  and 
how  much  ? 

24.  A  gentleman  gave  32  dollars  for  a  piece  of  cloth, 
and  13  dollars  for  having  it  made  into  a  suit  of  clothes. 
How  much  did  the  suit  cost  ? 

25.  A  tailor  gave  54  dollars  for  some  cloth  and  trim- 
mings; he  made  the  whole  into  clothes,  which  .he  sold 
for  72  dollars.     How  much  did  he  gain  by  the  work? 

26.  If  a  man,  having  50  dollars,  should  buy  a  barrel 
of  sugar  for  24  dollars,  and  a  barrel  of  molasses  for  13 
dollars,  how  many  dollars  would  he  have  left  ? 

27.  A  scholar  gave  55  cents  for  a  geography,  and  42 
cents  for  an  arithmetic.     What  did  he  give  for  both  .^ 

28.  Ellen  had  30  cents;  her  father  gave  16  more,  and 
her  mother  10  more;  she  then  bought  a  book  for  45  cents. 
How  many  cents  had  she  remaining  ? 

29.  James  had  38  cents,  and  his  father  gave  him  14 
more — William  had  33  cents,  and  his  father  gave  hhn 
19  more.     Which  boy  then  had  the  most  money  ? 

30.  Lucy  has  75  cents,  and  she  intends  buying  a  book, 
that  will  cost  63  cents.  How  much  money  has  she  more 
than  the  book  will  cost  ? 

31.  A  grocer  purchased  some  oranges  for  18  dollars, 
some  lemons  for  8  dollars,  some  raisins  for  4  dollars,  and 
some  figs  for  6  dollars;  and  then  sold  the  whole  for  44 
dollars.     Did  he  gain  or  lose  ? —  and  how  much  ? 

32.  A  company  marched  82  miles  in  three  days.  It 
marched  25  miles  the  first  day,  36  miles  the  second  day, . 
and  the  remainder  of  the  distance  the  third  day.  How 
many  miles  did  it  march  the  third  dav  ? 


M  ORAL    ARITHMETIC.  IV. 

CHAP.    IV. 

MULTIPLICATIOir. 

Note  to  Teachers,  In  tlie  second  and  third  sections  of  this  chapter,  the 
learners  are  required  to  find  the  products  of  factors  as  high  as  10  and  20. 
There  are  few  scholars,  who  will  easily  comuj  it  these  products  to  memory;  and 
it  will,  theiefore,  be  necessary  to  adopt  a  mental  process,  by  which  they  may 
leadiiy  be  found.     'J'he  following  example  will  show  the  process. 

Question.  How  many  are  8  times  16  1  Solution.  16  is  made  up  of  10 
and  6.  Eigli*  times  tea  are  eighty;  eight  times  six  are  forty-eight.  80  and 
48  ar€  128. 

Section  1. 

1.  On  the  Fourth  of  July,  George  and  Richard  went 
to  the  Celebration,  and  whenever  George  spent  1  cent, 
Richard  spent  7.  In  the  course  of  the  day,  George 
spent  6  cents.     How  many  did  Richard  spend  ? 

Solution,  Richard  must  have  spent  six  times  seven 
cents.     6  times  7  are  42. 

2.  If  4  yards  of  cloth  are  required  to  make  1  cloak, 
how  many  yards  are  required  to  make  5  cloaks  .'^ 

3.  If  1  boat  will  carry  5  men  across  the  river,  how 
many  men  will  3  boats  of  the  same  size  carry  ? 

4.  If  a  man  can  earn  8  dollars  in  one  week,  how  many 
dollars  can  he  earn  in  3  weeks  ^ 

5.  If  a  traveller  ride  6  miles  in  one  hour,  how  many 
miles  can  he  ride  in  9  hours  ? 

G.  In  a  garden,  there  are  5  rows  of  plum  trees;  8  trees 
in  each  row.     How  many  trees  are  there  in  the  garden  } 

7.  In  a  field  there  are  8  rows  of  apple  trees;  5  trees 
in  each  row.     How  many  trees  are  there  m  the  field  ? 

8.  There  were  7  boys,  who  gave  a  poor  man,  4  cents 
apiece.     How  many  cents  did  the  man  receive  ? 

9.  How  many  merit-marks  will  Susan  get  in  6  days, 
provided  she  gets  4  every  day  ? 

10  How  many  errors  will  Jane  make  in  8  days,  pro- 
vided she  makes  3  errors  every  day  ? 

11.  If  the  price  of  1  quart  of  nuts  b®  7  cents,  for  how 
many  cents  can  you  buy  7  quarts .'' 

12.  If  you  should  read  9  pages  every  day,  how  many 
pages  would  you  read  in  8  days  ? 


1.  2.  MULTIPLICATION.  25 

13.  How  many  dollars  must  I  pay  for  9  yards  of  cloth, 
that  is  worth  6  dollars  a  yard  ? 

14.  How  many  dollars  must  I  pay  for  9  barrels  of 
flour,  when  the  price  is  7  dollars  a  barrel  ? 

15.  If  4  bushels  of  wheat  are  required  for  1  barrel  of 
flour,  how  many  bushels  are  required  for  8  barrels  ? 

16.  What  is  the  cost  of  7  reams  of  letter  paper,  that 
is  sold  at  6  dollars  per  ream  ? 

17.  How  much  will  a  market-man  get  for  10  melons, 
if  he  sell  them  at  9  cents  apiece  ? 

18.  If  a  man  can  earn  9  dollars  in  one  month,  how 
many  dollars  can  he  earn  in  6  months  ? 

Section    2. 

1.  John  and  Henry  together,  caught  13  fishes  every 
morning;  and  it  always  happened,  that  John  caught  10 
of  them,  and  Henry  3.  Now  how  many  did  each  boy 
catch  in  6  mornings  ? — Then  how  many  did  they  both 
catch  in  6  mornings  ? 

2.  10  and  what  number  make  13  ?  How  many  are  6 
times  10?  6  times  3?  How  many  are  60  and  18? — 
Then  6  times  13  are  how  many  ? 

3.  If  the  diamond  in  a  ring  cost  10  dollars,  and  the 
ring  4  dollars,  what  does  the  diamond-ring  cost  ?  What 
would  5  diamonds  cost  ?  What  would  5  rings  cost  ? — 
Then  what  would  5  diamond-rings  cost  ? 

4.  10  and  what  number  make  14  ?  How  many  are  5 
times  10?  5  times  4?  How  many  are  50  and  20.^—^ 
Then  5  times  14  are  how  many  ? 

5.  A  Northern  hunter  received  a  bounty  of  10  dollars 
from  the  state,  and  5  from  the  county,  for  killing  a  wolf: 
how  much  did  he  receive  from  both  ?  How  much  would 
he  receive  from  the  state  for  killing  7  wolves  ?  How 
much  from  the  county  for  killing  7  ? — Then  how  much 
from  state  and  county  both  for  killing  7  ? 

6.  10  and  what  number  make  15  ?  How  many  are  7 
times  10  ?  7  times  5  ?  How  many  are  70  and  35  ? — 
Then  7  tim.es  15  are  how  many  ? 

7.  A  man  hired  a  horse  and  gig,  to  pay  10  cents  a 
mile  for  the  horse,  and  6  cents  a  mile  for  the  gig.     How 

c 


26 


ORAL    ARITHMETIC. 


IV. 


much  a  mile  did  he  pay  for  both  ?  How  much  for  the 
horse  8  miles  ?  How  much  for  the  gig  8  miles  ? — Then 
how  much  for  the  horse  and  gig,  both  8  miles  ? 

8.  10  and  what  number  make  16  ?  Haw  many  are  8 
times  10  ?  8  times  6  ?  How  many  are  80  and  48  ? — 
Then  8  times  16  are  how  many  ? 

9.  If  a  man  gather  10  barrels  of  apples,  and  a  boy  7 
barrels,  in  a  day,  how  many  barrels  do  they  both  gather? 
How  many  barrels  can  the  man  gather  in  9  days  ?  How 
many  barrels  can  the  boy  gather  in  9  days  ? — Then  how 
many  barrels  can  they  both  gather  in  9  days  ? 

10.  10  and  what  number  make  17  ?  How  many  are 
9  times  10  ?  9  times  7  ?  How  many  are  90  and  63  ? — 
Then  9  times  17  are  how  many  ? 

11.  If  a  man  eat  10  ounces  of  meat  in  a  day,  and  his 
wife  eat  8  ounces,  how  many  ounces  do  they  both  eat  in 
a  day  ?  How  many  ounces  will  the  man  eat  in  4  days  ? 
How  many  ounces  will  the  wife  eat  in  4  days  ? — Then 
how  many  ounces  will  they  both  eat  in  4  days  ? 

12.  10  and  what  number  make  18  .'^  How  many  are 
4  times  10  .^  4  times  8  ?  How  many  are  40  and  32  ? — 
Then  4  times  18  are  how  many? 

13.  If  a  company  of  soldiers  march  10  miles  in  the 
forenoon,  and  9  miles  in  the  afternoon,  how  many  miles 
do  they  march  in  a  day  ?  How  many  miles  would  they 
march  in  6  forenoons  ?  How  many  in  six  afternoons  ? — 
Then  how  many  miles  in  6  days  ? 

'  14.  10  and  what  number  make  19?  How  many  are 
6  times  10  ?  6  times  9  ?  How  many  are  60  and  54  ? — 
Then  6  times  19  are  how  many? 

15.  How  many 'are  6  times  10?  6  times  2? — Then 
how  many  are  6  times  12  ? 

16.  How  many  are  7  times  10?  7  times  3.^ — Then 
how  many  are  7  times  13  ? 

17.  How  many  are  8  times  10?  8  times  4? — Then 
how  many  are  8  times  14  ? 

18.  How  many  are  9  times  10?  9  times  d? — Then 
how  many  are  9  times  15  ? 

19.  How  many  are  10  times  10  ?  10  times  6  f — Then 
how  many  are  10  times  16  ? 


2.  3. 


MULTIPLICATION. 


27 

■Then 

-Then 

—Then 

-Then 


20.  How  many  are  3  times  10?     3  limes  7 
how  many  are  3  times  17  .'^ 

21.  How  many  are  2  times  10?     2  times  8 
how  many  are  2  times  IS  ? 

22.  How  many  are  5  times  10?     5  times  9 
how  m_any  are  5  times  19  ? 

23.  How  many  are  4  times  10?     4  times  9 
how  many  are  4  times  19  ? 

Section  3. 

1 .  Albert  spends  1 1  cents  every  month,  for  stationary. 
How  many  cents  will  he  spend  in  3  months  ? 

2.  How  many  are  2  times  11?  3  times  11?  4  times  11? 
5  times  11?  6  times  11?  7  times  11?  8  times  11?  9 
times  11?     10  times  11? 

3.  If  you  should  write  11  copy-lines  every  day,  how 
many  copy-lines  would  you  write  in  10  days  ? 

4.  If  a  pound  of  Malaga  raisins  cost  12  cents,  how 
many  cents  will  4  pounds  of  raisins  cost  ? 

5.  How  many  are  2  times  12?  3  times  12?  4  times  12? 
5  times  12?  6  times  12?  7  times  12?  8  times  12?  9 
times  12?     10  times  12? 

6.  If  you  should  read  12  verses  every  morning,  how 
many  verses  would  you  read  in  9  mornings  ? 

7.  Suppose  a  st^am  boat  will  go  13  miles  in  an  hour;— 
how  many  miles  will  it  go  in  5  hours  ? 

8.  Howmanyare2timesl3r  3timesl3?  4timesl3? 
5  times  13?  6  times  13?  7  times  13?  8  times  13?  9 
times  13?     10  times  13? 

9.  If  a  labourer  can  earn  13  dollars  in  a  month,  how 
many  dollars  can  he  earn  in  8  months  ? 

10.  If  it  take  14  men  to  navigate  one  ship,  how  many 
men  will  it  take  to  navigate  6  ships  ? 

11.  How  many  are  2  times  14?  3timesl4?  4timesl4? 
5  times  14?  6  times  14?  7  times  14?  8  times  14?  9 
times  14?     10  times  14? 

12.  If  a  pound  of  honey  be  worth  14  cents,  Jhow  many 
cents  are  7  pounds  of  honey  worth? 


28  ORAL    ARITHMETIC.  IV 

13.  If  a  carpenter  can  make  15  hat  boxes  in  a  day, 
how  many  hat  boxes  can  he  make  in  2  days  ? 

14.  Howmanyare2timesl5?  3timesl5?  4timesl5? 
5  times  15?  6  times  15?  7  times  15?  8  times  15?  9 
times  15?     10  times  15? 

15.  If  a  shoemaker  make  15  pairs  of  shoes  in  a  week, 
how  many  pairs  will  he  make  in  10  weeks  ? 

16.  How  much  would  a  man  earn  in  3  months,  pro- 
vided his  wages  were  16  dollars  a  month? 

17.  Howmanyare2timesl6?  3timesl6?  4timesl6? 
5  times  16?  6  times  16?  7  times  16?  8  times  16?  9 
times  16?     10  times  16? 

18.  How  much  would  a  man  spend  in  9  months,  if 
his  expenses  were  16  dollars  a  month? 

19.  If  17  barrels  of  flour  can  be  carried  on  one  wagon, 
how  many  barrels  may  be  carried  on  4  wagons  ? 

20.  How  many  are  2  times  17?  3timesl7?  4timesl7? 
5  times  17?  6  times  17?  7  times  17?  8  times  17?  9 
times  17?     10  times  17? 

21.  If  one  hogshead  of  molasses  be  worth  17  dollars, 
what  is  the  value  of  8  hogsheads  of  molasses  ? 

22.  If  5  men  should  pay  me  18  dollars  apiece,  how 
many  dollars  should  I  receive  from  them  all  ? 

23.  How  many  are  2  times  18?  3timesl8?  4timesl8? 
5  times  18?  6  times  18?  7  times  18?  8  times  18?  9 
times  18?     10  times  18? 

24.  If  I  should  pay  to  7  men,  18  dollars  apiece,  how 
many  dollars  should  I  pay  to  all  of  them? 

25.  How  much  would  a  farmer  get  for  3  cows,  if  he 
should  sell  them  for  19  dollars  apiece  ? 

26.  Howmanyare2timesl9?  3timesl9?  4timesl9? 
5  times  19?  6  times  19?  7  times  19?  8  times  19?  9 
times  19?  10  times  19? 

27.  If  you  answer  19  questions  at  every  recitation, 
how  many  would  you  answer  in  reciting  6  times? 

28.  Paper  is  generally  packed  in  reams,  of  20 quires 
each.     How  many  quires  are  there  in  4  reams? 


4.  MULTIPLICATION.  29 

29.  How  many  are  2  times  20?  3  times  20?  4  times  20? 
6  times  20  ?  6  times  20  ?  7  times  20  ?  8  times  20  ?  9 
times  20?     10  times  20? 

30.  If  a  trader  make  20  cents  on  every  pound  of  tea 
he  sells,  how  much  will  he  make  on  5  pounds. 

Section   4. 

1.  An  ounce  is  a  small  weight,  16  of  which  make  a 
pound.  How  many  ounces  of  tea  are  there  in  3  pounds 
and  9  ounces  of  tea? 

Solution.  In  1  pound  there  are  16  ounces,  and  in  3 
pounds  there  are  3  times  16  ounces.  3  times  16  ounces 
are  48  ounces.     48  ounces  and  9  ounces  are  57  ounces. 

2.  How  many  ounces  in  S-  pounds  and  4  ounces  ? 

3.  20  penny-weights  of  gold  make  1  ounce.  How 
m.any  penny-weights  in  4  ounces  and  13  penny- weights  ? 

4.  How  many  penny- weights  are  there  in  6  ounces 
and  9  penny-w^eights  ? 

5.  The  brewer  sells  his  beer  by  the  firkin,  and  a  fir- 
kin holds  as  much  as  9  gallon  measures.  How  many 
gallons  are  there  in  10  firkins  and  6  gallons  ? 

6.  In  7  firkins  and  3  gallons,  how  many  gallons  ? 

7.  40  rods,  measured  by  a  surveyor's  chain,  make  1 
furlong.      How  many  rods  in  3  furlongs  and  17  rods  ? 

8.  In  4  furlongs  and  8  rods,  how  many  rods  ? 

9.  There  are  12  months  in  a  year.  How  many 
months  are  there  in  6  years  and  6  months  ? 

10.  In  9  years  and  10  months,  how  many  months  ? 

11.  4  pecks  of  oats,  peas,  beans,  or  any  other  dry 
commodity,  make  1  bushel.  How  many  pecks  of  wheat 
are  there  in  6  bushels  and  3  pecks  ? 

12.  In  10  bushels  and  1  peck,  how  many  pecks  ? 

13.  12  pence,  in  English  money,  make  1  shilling. 
How  many  pence  are  .there  in  8  shillings  and  6  pence? 

14.  In  10  shillings  and  9  pence,  how  many  pence  ? 

15.  10  cents,  in  Federal  money,  make  1  dime.  How 
Hiany  cents  are  there  in  7  dimes  and  6  cents  ? 

16.  In  4  dimes  and  8  cents,  how  many  cents  ? 

17.  In  1  month  there  are  30  days.  How  many  days 
are  there  in  3  months  and  15  days  ^ 


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1  DIVISION.  31 

CHAP.    V. 
DIVISIOJV. 

Section  1. 

1.  A  lady  divided  15  peaches  among  some  little  girls, 
giving  3  to  each  girl.     How  many  girls  were  there  ? 

Solution.  As  many  times  as  3  peaches  are  contained 
in  15  peaches,  so  many  girls  there  were. 

2.  If  you  had  16  cents  to  lay  out  in  pencils,  and  the 
price  of  the  pencils  were  4  cents  apiece,  how  many 
could  you  buy  for  all  the  money  ? 

3.  How  many  times  is  4  contained  in  16  ? 

4.  If  4  horses  are  required  to  draw  one  wagon,  how 
many  wagons  might  be  drawn  by  20  horses  ? 

5.  How  many  times  4  in  20  ?     How  many  are  5  times  i? 

6.  If  a  man  can  travel  4  miles  in  one  hour,  how  many 
hours  will  it  take  him  to  travel  12  miles  ? 

7.  How  many  times  4  in  12  ?     Hotv  many  are  3  times  4? 

8.  How  many  yards  of  broadcloth,  that  is  sold  at  7 
dollars  a  yard,  can  be  purchased  for  14  dollars  ? 

9.  How  many  times  7  in  14  ?     How  muny  are  2  times  7? 

10.  How  many  lead  pencils  could  you  buy  for  18 
cents,  if  they  were  sold  at  6  cents  apiece  ? 

11.  How  many  times  6  in  18  ?     Hoiv  many  are  S  times  6? 

12.  In  an  orchard  there  are  35  trees,  standing  in  rows, 
7  trees  in  a  row.     How  many  rows  are  there  ? 

13.  How  many  times  7  in  35?     Howmanyare  5times7? 

14.  A  man  bought  sheep  at  4  dollars  apiece,  and  paid 
for  them  all,  24  dollars.     How  many  did  he  buy  ? 

15.  How  many  times  4  in  24  ?     How  many  are  Q  times  A7 

16.  A  gardener  set  out  30  peach  trees,  in  rows,  put- 
ting 5  trees  in  a  row.     How  many  rows  were  there  } 

17    How  many  times  5  in  30.^     How  m^ny  are  6  times  5? 
18.  A  farmer  got  36  dollars  for  some  sheep,  that  he 
Bold  at  6  dollars  apiece.     How  many  were  there  ? 
19    How  many  times  6  in  36  }     How  many  are  6  times  6? 


32  ORAL    ARITHMETIC.  V\ 

20.  A  trader  wishes  to  pack  56  hats  in  boxes,  putting 
8  hats  in  a  box: — how  many  boxes  are  wanted  ? 

21.  How  many  times  8  in  56  ?     How  many  are  7  times  S? 

22.  How  many  dozen  of  eggs  can  you  buy  for  63 
cents,  when  they  are  sold  at  9  cents  a  dozen? 

23.  How  many  times  9  in  63  .'^     How  many  artV  times  9? 
•24.  If  an  orange  be  worth  6  cents,  and  a  hme  1  cent, 

then  how  many  oranges  are  60  limes  worth  ? 

25.  How  many  times  6  in  60  ?     How  many  ar  elO  times  G? 

26.  If  one  silk  bonnet  be  Worth  7  handkerchiefs,  how 
many  bonnets  ar-e  56  handkerchiefs  worth  ? 

27.  How  many  times  7  in  56  ?     How  many  are  S  times  7? 

28.  If  I  give  a  barrel  of  flour  for  4  bushels  of  wheat, 
how  many  barrels  must  I  give  for  36  bushels  ? 

29.  How  many  times  4  in  36.^      How  many  are  9  times ^'t 

30.  If  a  man  can  build  S^rods  of  fence  in  a  day,  how 
many  days  will  it  take  him  to  build  72  rods  ? 

31.  How  many  times  8  in  72  ?     Eoicf  many  ar e  9  times  S? 

32.  If  5  bushels  of  wheat  will  pay  for  a  yard  of  broad 
cloth,  how  many  yards  will  45.bus]iels  pay  for? 

33.  How  many  times  5  in  45?     How  many  are  9  times  ^? 
.34.  Lafayette  was  42  days  on  iiis  passa  ge  from  Toulon 

to  New  York.     How  many  weeks  !was  hi,  ^  passage  ? 
35.   How  many  times  7  in  42?'    Howma\  iyare6times7? 

Section  2. 
1.   Suppose  2  men  have  8  biscuit  to  diviv  ^^  equally 
between  them; — how  many  must  each.m^n  tak  ^^ ^ 

Observution,  There  are  2  men  to  share  the  biscuit, 
and  if  there  were  only  2  biscuit  to  her  divided,  ^  ^ 
man  would  take  1  biscuit.  Therefore,!  man  wii  ^  ^^^^ 
take  1  biscuit, of  every  2  bisctiit. 

Solution,  As  many  times  as  2  is  contained  in  t  '•>  ^^ 
many  biscuit  must  each  man  take.  2  is  contained  it  ■  ^^ 
4  jtimes. 

^.  Jf  12  dollars  be  divided  equally  betwe.en  '2  mei  h 
\y  many  dollars  does  each  man  receive? 
\    Jf  18  chestnuts  should  be  divii^ed  equally  between 
^g,  how  many  would  each  jbw  ^^give  ? 


2.  DIVISION  33 

4.  A  tenant  cultivated  a  piece  of  corn,  agreeing  to  give 
the  owner  of  the  land  1  bushel  of  every  2  bushels  that 
he  might  raise.  He  raised  22  bushels.  How  many- 
bushels  should  the  owner  of  the  land  receive  ? 

5.  Suppose  3  boys  have  12  oranges  to  divide  equally 
between  them; — how  many  must  each  boy  take  } 

Observation,  If  the  3  boys  had  only  3  oranges  to  di- 
vide, each  boy  would  take  I  orange; — if  they  had  2  times 
3  oranges,  each  boy  would  take  2  times  1  orange: — and 
thus,  each  boy  will  take  as  many  times  1  orange,  as  there 
are  threes  in  the  number  to  be  divided. 

Solution.  As  many  times  as  3  is  contained  in  12,  so 
many  oranges  must  each  boy  take.     3  in  12,  4  times, 

6.  If  15  biscuit  be  divided  equally  between  3  men, 
how  many  biscuit  does  1  man  receive.^ 

7.  3  careless  boys  must  pay  24  cents  for  breaking  a 
square  of  glass.     What  must  each  boy  pay.'^ 

8.  A  man  set  out  27  trees,  in  3  rows;  an  equal  num- 
ber in  each  row.     How  many  were  there  in  one  row  ? 

9.  4  boys  have  12  oranges  to  divide  equally  between 
them.     How  many  will  each  boy  receive  ? 

Observation.  If  the  4  boys  had  only  4  oranges  to  di- 
vide, then  each  boy  would  receive  1  orange.  Therefore 
each  boy  must  receive  1  orange  of  every  4  oranges  that 
there  are  in  the  number  to  be  divided. 

Solution.  As  many  times  as  4  is  contained  in  12,  so 
many  oranges  will  each  boy  receive.     4  in  12,  3  times. 

10.  I  have  32  minutes  to  spend  on  4  lessons.  How 
many  minutes  can  I, spend  on  each  lesson? 

11.  Suppose  I  wish  to  give* 28  quills  to  4  boys; — how 
many  must  I  give  to  each  boy? 

12.  4  men  received  20  dollars  for  doing  a  piece  of 
work.     How  much  was  each  man's  share  ? 

13.  A  fisherman  hired  a  boat,  agreeing  to  give  the 
owner,  1  fish  of  every  5,  that  he  might  catch:  he  caught 
20.     How  many  should  he  give  the  owner  ? 

14.  If  35  pounds  of  beef  be  divided  among  5  soldiers, 
how  many  pounds  does  each  soldier  receive  ? 

15.  5  men  have  agreed  to  pay  equal  shares  of  50  dol- 
lars     How  many  dollars  must  one  man  g^y? 


34  ORAL    ARITHMETIC.  V. 

16.  Charles  was  one  of  6  boys,  who  owned  together 
42  books.  They  divided  the  books,  and  Charles  receiv- 
ed 1  book  of  every  6  books.     How  many  did  he  receive.'* 

17.  If  24  books  should  be  divided  equally* among  6 
boys,  how  many  would  each  boy  receive  ? 

18.  6  men  have  agreed  to  pay  36  dollars  in  equal 
shares.     How  many  dollars  must  each  man  pay.^ 

19.  Edward  is  one  of  7  boys,  who  are  to  have  23 
peaches  divided  equally  among  them.  How  many  will 
Edward  receive  for  his  share  ? 

20.  If  7  writing-books  be  made  of  42  sheets  of  paper 
how  many  sheets  are  there  in  each  book  ? 

21.  8  boys  owned  together  72  quills;  and,  in  order  to 
share  them  equally,  each  boy  took  1  quill  from  every  8 
quills  in  the  number.     How  many  did  each  boy  take  ? 

22.  I  have  48  dollars  to  divide  among  8  men.  How 
many  dollars  must  I  give  to  one  man .''  « 

23.  9  men  shared  45  bushels  of  corn  among  them, 
each  man  taking  for  his  share,  1  bushel  of  every  9  bush- 
els.    How  many  bushels  did  each  man  take  ? 

24.  9  persons  have  agreed  to  make  up  a  purse  of  72 
dollars.     How  many  dollars  must  each  one  put  in  ? 

25.  10  sailors  are  to  receive  90  dollars  for  retaking 
their  ship.     How  much  will  each  sailor  receive  ? 

Section  3. 

1.  Charles  has  25  cents,  which  he  has  engaged  to 
appropriate  as  follows.  Whenever  he  blots  his  writing- 
book,  he  is  to  lay  it  aside,  and  pay  6  cents  for  a  new  one. 
Now  how  many  books  can  he  pay  for;  and  how  many 
cents  will  he  have  remaining,  after  his  number  becomes 
too  small  to  buy  another  book. 

Observation.  If  his  whole  number  of  cents  were  30, 
he  could  then  pay  for  five  books ;  because,  6  cents  are  con- 
tained in  30  cents,  5  times.  Again,  if  his  whole  number 
of  cents  were  only  24,  he  could  then  pay  for  4  books; 
because,  6  cents  are  contained  in  24  cents,  4  times. 

2.  Suppose  it  takes  8  buttons  to  trim  a  vest: — how 
many  vests  can  the  tailor,  who  has  only  34  buttons,  trim* 
and  what  number  of  buttons  will  he  have  remaining  ? 


3.  DIVISION.  35 

3.  How  many  times  8  in  34;  and  how  many  over  ? 

4.  How  many  glass  tumblers,  at  10  cents  apiece,  can 
a  woman  that  has  only  53  cents,  buy;  and  how  many 
cents  will  she  have  remaining  ? 

Solution,  As  many  times  as  10  is  contained  in  53,  so 
many  tumblers  she  can  buy.  10  is  contained  in  53,  5 
times,  and  there  is  three  over. 

5.  How  many  kegs,  that  will  hold  7  gallons  apiece, 
may  be  filled  from  a  cask  of  wine  containing  46  gallons; 
and  how  many  gallons  will  remain  in  the  cask  ? 

6.  How  many  times  7  in  46;  and  how  many  over.'* 

7.  A  hat-maker  has  53  hats  finisiied;  and,  in  order  to 
send  them  to  market,  he  must  pack  them  in  boxes,  that 
will  hold  8  hats  apiece.  How  many  full  boxes  can  he 
send;  and  how  many  hats  will  remain  on  hand  ? 

8.  How  many  times  8  in  53;  and  how  many  over  ? 

9.  A  trader  has  69  dollars  with  which  he  wishes  to 
purchase  hats.  If  he  should  pay  7  dollars  apiece  for  the 
hats,  how  many  could  he  purchase;  and  how  many  dol- 
lars would  he  have  remaining  ? 

10.  How  many  times  7  in  69;  and  how  many  over  -* 

11.  If  4  yards  of  cloth  will  make  1  cloak,  how  many 
cloaks  can  be  made  from  a  piece  of  cloth  containing  38 
yards;  and  how  many  yards  will  there  be  over  .'^ 

12.  How  many  times  4  in  38;  and  hew  many  over  .'^ 

13.  How  many  times  is  4  contained  in  29;  and  how 
many  over  r      How  many  are  7  times  4,  and  1  more? 

14.  How  many  times  is  6  contained  in  53;  and  how 
many  over  }      How  many  are  8  times  6,  and  5  more  ? 

15.  How  many  times  is  9  contained  in  57;  and  how 
many  over  }     How  many  are  6  times  9,  and  3  more  ? 

16.  How  many  times  is  7  contained  in  68;  and  how 
many  over  }     How  many  m-e  9  times  7,  and  5  more  ? 

17.  How  many  times  is  5  contained  in  49;  and  how 
many  over  :     How  many  are  9  times  5,  and  4  mx)re  ? 

18.  How  many  times  is  3  contained  in  2Q'.,  and  Jbow 
many  over  }     How  many  are  8  times  3,  and  2  more  ? 

19.  In  15,  how  many  times  4;  and  how  many  ovqy? 
In  17  ?     In  26  ?     In  33  ?     In  27  }     In  42  .? 


36  ORAL    ARITHMETIC.  V. 

20.  In  27,  how  many  times  5;  and  how  many  over? 
In  29?     In  36?     In  32  ?     In  44  ?     In  48  ? 

21.  In  28,  how  many  times  6;  and  how  many  over? 
In  37  ?     In  38  ?     In  46  ?     In  49  ?     In  22  ? 

22.  In  30,  how  many  times  7;  and  how  many  over  ? 
In  36?     In  43?     In  48  ?     In  51  ?     In  59  ? 

23.  In  28,  how  many  times  8;  and  how  many  over? 
Tn  35  ?     In  46  ?     In  52  ?     In  61  ?     In  75  ? 

24.  In  31,  how  many  times  9;  and  how  many  over  ? 
tn  34  ?     In  42  ?     In  50  ?     In  67  ?     In  70  ? 

25.  A  gallon  measure,  used  for  jneasuring  wine,  beer, 
milk,  &c.  will  contain  as  much  as  4  quart  measures. 
Suppose  I  have  15  quart  measures  full  of  water; — how 
many  gallon  measures  can  I  fill  from  them;  and  how 
many  quarts  will  there  be  over  ? 

26.  How  many  gallons  are  there  in  34  quarts  ? 

27.  3  feet,  measured  on  a  line,  are  the  same  as  1  yard. 
How  many  yards  are  there  in  29  feet  ? 

28.  How  many  yards  are  there  in  17  feet  ? 

29.  60  minutes,  by  the  clock,  make  1  hour.  How 
many  hours  are  there  in  128  minutes  ? 

30.  12  inches,  on  the  carpenter's  rule,  make  1  foot. 
How  many  feet  long  is  a  board,  that  is  65  inches  long  ? 

31.  How  many  feet  are  there  in  38  inches  ? 

32.  8  drams  of  medicine,  weighed  by  the  apothecary, 
are  the  same  as  1  ounce  of  medicine.  How  many  ounces 
are  there  in  46  drams  ? 

33.  How  many  ounces  are  there  in  30  drams  ? 

34.  9  square  feet,  measured  upon  the  floor,  make  1 
square  yard.  How  many  square  yards  are  there  in  20 
square  feet? 

35.  How  many  square  yards  in  51  square  feet? 

36    7  days  are  1  week.     How  many  weeks  in  33  days  ? 

37.  How  many  weeks  are  there  in  52  days  ? 

38.  12pence,  in  Enghsh money,  make  Ishilhng.  How 
.^any  shillings  are  there  in  69  pence  ? 

39.  How  many  shillings  are  there  in  46  pence  ? 

40.  10  cents,  in  Federal  money,  make  1  dime.  How 
many  dimes  are  there  in  48  cents  ? 

41.  How  many  dimes  are  there  in  95  cents  ? 


4.  MULTIPLICATION    AND    DIVISION.  37 

Sectiojv   4. 

CORRESPONDENT    EXAMPLES. 

1.  How  much  will  5  pounds  of  dates  cost,  when  the 
price  of  them  is  7  cents  a  pound  ? 

Solution,  If  one  pound  cost  7  cents,  5  pounds  will 
cost  5  times  7  cents.     5  times  7  are  35. 

2.  How  many  pounds  of  dates  can  you  buy  for  35  cents; 
the  price  being  7  cents  a  pound  ? 

Solution.  I  can  buy  as  many  pounds  of  dates  as  7  is 
contained  times  in  35.     7  in  35,  5  times. 

3.  How  many  quarts  of  wine  are  there  in  10  gallons; 
there  being  4  quarts  in  one  gallon  ? 

4.  How  many  gallons  of  wine  are  there  in  40  quarts; 
every  4  quarts  making  one  gallon  ? 

5.  How  many  pence  are  there  in  5  shillings;  there 
being  12  pence  in  1  shiUing  ? 

6.  How  many  shillings  are  there  in  60  pence;  there 
being  12  pence  in  1  shiihng  ? 

7.  In  1  penny  there  are  4  farthings.  How  many  far- 
things in  9  pence  and  3  farthings  ? 

8.  How  many  pence  are  there  in  39  farthings; — and 
how  many  farthings  are  there  over  ? 

9.  A  gentleman  went  on  a  journey  of  9  days,  and  paid 
for  his  expenses,  4  dollars  per  day.  How  much  were 
his  expenses  during  the  whole  journey  ? 

10.  A  gentleman,  who  had  been  away  on  a  journey 
for  9  days,  found  on  his  return,  that  he  had  spent  36  dol- 
lars.    How  much  did  he  spend  a  day  ? 

Suggestion.     9  dollars  would  allov/  him  1  dollar  a  day. 

11.  If  6  bushels  of  onions  grow  upon  1  square  rod  of 
ground,  how  many  bushels  will  grow  upon  10  rods  ? 

12.  A  man  raised  60  bushels  of  onions  upon  10  rods 
of  ground.     How  many  bushels  grew  upon  1  rod } 

13.  If  a  ship  sail  7  miles  an  hour,  how  many  miles 
will  she  sail  in  7  hours  ? 

14.  If  a  ship  sail  49  miles  in  7  hours,  how  many  mileg 
does  she  sail  in  1  hour  ':' 


38  ORAL    ARITHMETIC.  V. 

Section   5. 

CONNECTED    OPERATIONS. 

1.  A  market  man  sold  10  pounds  of  cheese  at  8  cents 
a  pound,  and  received  his  pay  in  sugar  at  10  cents  a 
pound.     How  many  pounds  of  sugar  did  he  receive  ? 

Solution.  The  price  of  1  pound  of  cheese  being  8 
cents,  the  price  of  10  pounds  is  10  times  8  cents,  or  80 

cents. 80  cents  will  pay  for  as  many  pounds  of  sugar, 

as  there  are  times  10  in  SO.     10  in  80,  8  times. 

2.  In  10  times  8,  how  many  times  10? 
Solution.     10  times  8  are  80. 10  in  80,  8  times. 

3.  4  coaches  went  from  Baltimore  to  Washington, 
each  carrying  6  boys:  the  same  boys  returned,  riding  8 
in  a  coach.     In  how  many  coaches  did  they  return  ^ 

4.  In  4  times  6,  how  many  times  8  ? 

5.  How  maoy  boxes  of  raisins  at  6  dollars  a  box,  will 
pay  for  4  kegs  of  tobacco  at  9  dollars  a  keg  ? 

6.  In  4  times  9,  how  many  times  6  ? 

7.  How  many  sheep  worth  4  dollars  a  head,  must  be 
given  for  6  tons  of  hay,  worth  8  dollars  a  ton  ? 

8.  In  6  times  8,  how  many  times  4  ? 

'    9.  How  many  reams  of  paper  at  3  dollars  a  ream,  will 
pay  for  5  dozen  of  books  at  6  dollars  a  dozen  ? 

10.  In  5  times  6,  how  many  times  3  ? 

11.  A  hunter,  in  Michigan,  sold  7  pelts  at  5  dollars  a 
pelt,  agreeing  to  take  his  pay  in  muskets  at  S  dollars  apiece 
The  purchaser  counted  out  as  many  muskets  as  the  pelts 
would  pay  for,  and  finding  there  was  still  a  balance  due 
to  the  hunter,  he  paid  this  in  money.  How  many  mus- 
kets and  how  much  money  did  the  hunter  receive  ? 

12.  In  7  times  5  how  many  times  8;  how  many  over 

13.  In  3  times  4  how  many  times  5;  how  many  over 

14.  In  8  times  5  how  many  times  7;  how  many  over 

15.  In  7  times  7  how  many  times  5;  how  many  over 

16.  In  5  times  6  how  many  times  4;  how  many  over 

17.  In  9  times  3  how  many  times  6;  how  many  over 

18.  In  4  times  8  how  many  times  9;  how  many  over 

19.  In  6  times  9  how  many  times  8;  how  many  over 


5.  6.  MISCELLANEOUS    EXAMPLES.  39 

Note  to  Teachers.  In  performing  oral  solutions  which  involve  several  of»- 
erations,  it  will  be  found  convenient,  although  not  absolutely  necessary,  to  use 
the  terms  j)/ttsand  minus.  These  terms,  however,  should  not  be  adopted  until 
they  are  perfectly  understood;  and  the  duty  of  explaining  them  is  here  confided 
to  the  Teacher.  It  will  not  suffice,  merely  to  tell  the  learner,  that  "plus  means 
more  J  and  minus  means  less,''  for  the  idiom  of  our  language  does  not  allow  him 
to  use  more  and  less,  as  he  is  called  upon  to  use  plus  and  minus.  The  ex- 
pressions, "  7  more  5  is  12,"  and,  **  10  less  4  is  6,'*  although  true,  are  not 
very  likely  to  meet  the  understanding  of  a  young  scholar.  Perhaps  it  will  be 
found  necessary  to  resort  to  illustrations  like  the  following. 

Place  8  books  in  a  pile  before  you,  and  say, — "  Here  is  a  pile  of  8  books, 
and  I  shall  make  the  number  of  books  in  the  pile  3  more."  I'hen,  placing  3 
additional  books  upon  the  pile,  say, — "  8  books  plus  3  books  are  11  books." 
Again  proceed  to  say,—"  I  shall  now  make  the  number  2  less."  Then,  taking 
off  2  books,  say, — "  11  books,  minus  2  books  are  9  books." 

Section  6. 

MISCELLANEOUS    EXAMPLES. 

1.  How  many  days  in  the  three  Summer  months; — ' 
there  being  30  in  June,  31  in  July,  and  31  in  August  ? 

Solution,  30  days  plus  31  days  are  61  days;  61  days 
plus  31  days  are  92  days. 

2.  There  were  42  gallons  of  wine  in  a  cask;  but,  the 
cask  not  being  tight,  7  gallons  have  leaked  out.  How 
many  gallons  still  remain  in  the  cask  ? 

Solution.     42  minus  7  is  35.     Answer.  35  gallons. 

3.  A  boy  that  had  97  cents,  paid  62  cents  for  a  book, 
18  cents  for  a  morocco  wallet,  and  6  cents  for  a  pencil. 
How  many  cents  had  he  remaining  ? 

Solution.  62  cents  (for  book),  plus  IS  cents  (for 
wallet), is  80  cents;  80  cents  plus  6  cents  (for  pencil), 
is  86  cents.     97  cents  minus  86  cents  is  11  cents. 

4.  A  black-smith  bought  9  tons  of  coal,  at  4  dollars 
per  ton,  and  gave  3  dollars  for  having  it  drawn  to  his 
shop.     How  much  did  the  coal  cost  him  ? 

Solution,  If  the  price  of  1  ton  was  4  dollars,  the  price 
of  9  tons  was  9  times  4  dollars,  or  36  dollars;  36  dollars 
plus  3  dollars  (for  having  it  drawn)  is  39  dollars. 

5.  A  schoolmaster  laid  out  96  cents  in  writing-books, 
at  8  cents  apiece,  and  then  gave  away  5  of  them.  How 
many  books  had  he  remaining  ? 

Solution.  He  bought  as  many  books,  as  8  is  contained 
times  in  96,  which  is  12.  Then,  12  books  minus  5  books, 
(that  he  gave  away),  are  7  books. 


40  ORAL    ARITHMETIC.  V 

6.  A  trader  gave  16  dollars  for  a  keg  of  tobacco,  and 
after  selling  it,  found  he  had  gained  6  dollars.  For  how 
much  did  he  sell  it  ? 

7.  A  rich  farmer  in  Vermont  had  a  flock  of  100  sheep; 
they  went  upon  a  mountain,  and  the  wolves  destroyed  IB 
of  them.     How  many  sheep  had  he  remaining  ? 

8.  If  a  man  spend  4  dollars  in  a  week,  how  many  dol 
lars  will  he  spend  in  9  weeks  ? 

9.  How  many  dozen  of  eggs  can  you  buy  for  64  cents, 
when  the  price  of  them  is  8  cents  per  dozen  ? 

10.  A  trader  gave  48  dollars  for  7  barrels  of  flour,  and 
sold  it  for  6  dollars  a  barrel.     What  did  he  lose  ? 

1 1 .  How  many  weeks  are  there  in  35  days  ? 

12.  Four  men  made  up  a  purse  of  40  dollars,  for  a 
charitable  purpose.  The  first  man  put  in  9  dollars,  the 
second  12  dollars,  and  the  third  7  dollars; — how  much 
did  the  fourth  man  put  in  ^ 

13.  A  sum  of  money  was  divided  equally  among  8 
sailors;  and  Jack,  who  was  one  of  the  number,  received 
for  his  share,  15  dollars.     What  was  the  sum  divided? 

14.  How  many  days  are  there  in  13  weeks  and  5  days.^ 

15.  A  trader  bought  3  reams  of  paper,  at  5  dollars  per 
ream,  and  7  maps,  at  5  dollars  apiece.  How  much  did 
he  give  for  the  whole  ? 

16.  If  49  bushels  of  corn  should  be  divided  equally 
among  7  men,  how  much  would  one  man  receive  ? 

Solution.  As-  many  times  as  7  is  contained  in  49,  so 
many  bushels  would  one  man  receive. 

17.  If  45  dollars  be  divided  equally  between  5  men, 
how  many  dollars  does  each  maa  receive  ? 

18.  A  man  bought  a  turkey  weighing  10  pounds,  for  8 
cents  a  pound,  and  then  sold  it  for  3  cents  a  pound  more 
than  he  gave.     For  how  much  did  he  sell  it  ? 

19.  Charles  had  25  cents;  his  father  gave  him  4  mel- 
ons, which  he  sold  for  6  cents  apiece;  he  then  paid  12 
cents  for  a  book.     How  many  cents  had  he  left  ? 

20.  If  a  man  earn  6  dollars  a  week,  how  many  weeks 
will  it  take  him  to  earn  48  dollars  ? 

21.  Geographers  consider  the  United  States  in  four 
classes.     There  are  6  Eastern  states,  4  Middle  states, 


6.  MISCELLANEOUS   EXAMPLES.  41 

8  Southern  states,  and  6  Western  states.     How  many 
states  are  there  in  the  Union  ? 

22.  A  merchant  paid  43  dollars  for  some  iron,  and  sold 
it  for  35  dollars.     How  many  dollars  did  he  lose  ? 

23.  A  man  paid  78  dollars  for  a  piece  of  land,  and  16 
dollars  for*having  it  fenced;  and  he  then  sold  it  for  100 
dollars.     Did  he  gain  or  lose; — and  how  much  ? 

24.  A  cabinet-maker  sold  6  tables,  at  14  dollars  apiece. 
How  many  dollars  did  he  receive  ? 

25.  If  I  bay  10  yards  of  cloth,  at  7  dollars  a  yard,  how 
many  five-dollar  bills  must  I  pay  for  it  ? 

26.  How  many  boxes  of  strawberries  can  you  buy  for 
36  cents,  when  they  are  sold  at  9  cents  a  box  ? 

27.  Suppose  a  trader,  who  has  12  barrels  of  flour  on 
hand,  should  lay  out  35  dollars  in  buying  more  flour,  at 
5  dollars  a  barrel;  how  many  barrels  would  he  have  ? 

28.  If  I  pay  19  dollars  to  one  man,  13  to  another, 
and  31  to  another,  how  many  dollars  do  I  pay  out  ? 

29.  If  a  laborer  can  earn  7  dollars  in  a  week,  how 
many  weeks  will  he  be  in  earning  42  dollars  ? 

30.  How  many  hats,  that  are  sold  at  6  dollars  apiece, 
can  a  man  who  has  50  dollars  pay  for; — and  how  many 
dollars  will  he  have  remaining  ? 

31.  If  you  should  perform  19  examples  in  arithmetic, 
every  day,  how  many  would  you  perform  in  6  days  ^ 

32.  Samuel  Moderate  earns  7  dollars  a  month,  and 
John  Smart  earns  15  dollars  a  month.  How  much  more 
will  John  earn  than  Samuel,  in  6  months  ? 

33.  If  1  man  do  1  day's  work  in  1  day,  how  many  men 
will  it  take  to  perform  7  days'  work,  in  1  day  ? 

34.  If  4  men  will  perform  4  days'  work  in  1  day,  how 
many  days'  work  will  4  men  perform  in  9  days  ^ 

35.  How  many  days  will  it  take  4  men  to  dig  a  cellar, 
that  1  man  would  be  36  days  in  digging  ? 

36.  How  many  days  will  it  take  7  men  to  clear  a  piece 
of  wood-land,  that  28  men  can  clear  in  one  day  ? 

37.  How  many  men  will  it  take  to  perform  as  much 
work  in  1  day,  as  1 1  men  can  perform  in  6  days  ? 

38.  How  many  days  will  it  take  4  men  to  perform  the 
same  work,  that  12  men  can  perform  in  3  days  ? 


42  ORAL    ARITHMETIC.  V 

39.  A  trader  has  three  bundles  of  bank  notes; — 23 
dollars  in  one  bundle,  15  dollars  in  another,  and  34  dol- 
lars in  another;  but  in  one  of  the  bundles  there  is  a 
note  of  5  dollars,  which  is  counterfeit.  How  many  dol- 
lars of  good  money  has  he  ? 

40.  Stephen  has  lost  30  cents,  and  has  found  10  cents; 
he  now  has  18  cents.     How  much  had  he  at  first  ? 

41.  A  farmer  went  to  the  city  with  8  barrels  of  cider, 
which  he  sold  at  4  dollars  a  barrel.     He  then  purchased 

3  hogsheads  of  salt,  at  3  dollars  per  hogshead,  and  paid  an 
old  debt  of  12  dollars.  How  many  dollars  had  he  to 
carry  home  ? 

42.  If  I  pay  3  dollars  apiece  for  7  umbrellas,  and  6 
dollars  apiece  for  6  hats,  for  how  many  dollars  must  I 
sell  the  whole,  in  order  to  gain  7  dollars  ? 

43.  A  man  borrowed  75  dollars,  and  the  next  day  paid 
all  but  14  dollars  of  it.     How  much  did  he  pay  ? 

44.  A.  spent  5  dollars  as  often  as  B.  spent  3  dollars. 
How  much  did  A.  spend,  while  B.  spent  27  dollars  ? 

Solution,  As  many  times  as  3  dolkrs  are  contained 
in  27  dollars,  so  many  times  5  dollars  did  A.  spend.  3 
dollars  are  contained  in  27  dollars  9  times;  therefore  A. 
spent  9  times  5  dollars. 

45.  A  man  and  a  boy  were  gathering  corn; — the  man 
gathered  7  rows,  in  the  same  time  that  the  boy  gathered 

4  rows.  How  many  rows  would  the  man  gather,  while 
the  boy  was  gathering  32  rows  ? 

46.  A  man  and  a  boy  were  digging  potatoes; — the 
man  dug  11  bushels  in  the  same  time  that  the  boy  dug  G 
bushels.  How  many  bushels  would  the  boy  dig,  while 
the  man  was  digging  55  bushels  ? 

47.  Suppose  butter  to  be  worth  12  cents  a  pound,  and 
tea  42  cents  a  pound; — how  many  pounds  of  butter  must 
be  given  for  2  pounds  of  tea  ? 

'48.  A  farmer  sold  2  cows  at  23  dollars  apiece,  and  9 
sheep  at  5  dollars  apiece;  he  received  in  payment,  3 
ploughs  at  8  dollars  apiece,  and  the  rest  in  money.  How 
much  money  did  he  receive  ? 

49.  4  boys  found  a  purse  containing  29  dollars.  They 
paid  2  dollars  for  advertising  it;  and,  as  no  owner  ap- 


(5.  MISCELLANEOUS   EXAMPLES.  43 

peared,  they  agreed  to  take  6  dollars  apiece  to  themselves, 
and  give  the  remainder  to  a  poor  woman.  How  much 
was  there  remaining  for  the  woman  ? 

50.  What  sum  of  money  must  be  divided  among  16 
men,  in  order  that  one  man  shall  receive  4  dollars  ? 

51.  Two  classes  are  studying  arithmetic.  The  first 
class  is  81  examples  in  advance  of  the  second;  the  sec- 
ond performs  40  examples  in  a  day,  and  the  first,  31.  In 
how  many  days  will  the  second  overtake  the  first  ? 

52.  A  lady  paid  6  dollars  for  silk,  9  dollars  for  cam- 
bric, 7  dollars  for  linen,  and  then  had  13  dollars  remain- 
ing.    How  many  dollars  had  she  at  first  ? 

53.  How  many  barrels  of  flour,  at  6  dollars  per  barrel, 
can  the  baker  who  has  45  dollars,  purchase;  and  how 
many  dollars  will  he  have  remaining  ? 

54.  A  trader,  that  has  48  dollars,  wishes  to  buy  all  the 
boots  he  can  pay  for,  at  5  dollars  a  pair,  and  then  lay  out 
the  remainder  of  his  money  in  shoes,  at  1  dollar  a  pair. 
How  many  pairs  of  boots,  and  of  shoes,  must  he  buy  ? 

55.  What  sum  of  money  must  be  divided  among  IS 
men,  in  order  that  one  man  shall  receive  7  dollars  ? 

56.  Three  men  made  up  a  purse  of  40  dollars.  The 
first  man  put  in  6  dollars,  and  the  second  3  times  as  much 
as  the  first.     How  much  did  the  third  put  in  ^ 

57.  Eliza  gave  a  poor  woman  4  cents,  Augusta  gave 
her  3  times  as  much  as  EHza,  and  Lucy  3  times  as  much 
as  Augusta.     How  much  did  the  woman  receive  ? 

58.  If  a  man  dig  30  bushels  of  potatoes  in  a  day,  and  a 
boy  13  bushels,  how  many  bushels  will  they  both  dig  in  a 
day  ?     How  many  will  they  both  dig  in  3  days  ? 

59.  A  farmer  purchased  15  sheep; — he  sold  8  of  them 
at  4  dollars  apiece,  and  the  remainder  at  3  dollars  apiece; 
and  ihen  found  that  he  had  gained  7  dollars.  How  much 
did  be  give  for  the  sheep  ? 


44 


ORAL    ARITHMETIC 


VI 


CHAP.  VI. 


FRACTIOI^^S. 

Section  1. 

Note  to  Teachers.  The  subsequent  progress  of  the  learner,  will  depend 
much  on  a  proper  conception  of  the  division  of  unity,  and  a  correct  application 
of  the  nomenclature  of  fractions.  Therefore,  this  section,  however  simple  it  may 
appear,  should  not  be  slighted.     It  should  be  recited  with  the  books  closed. 

The  picture  of  a  board. 

This  board,  as  it  is  presented  above,  is  a  whole  thing. 
The  same  board  appears  hereafter  divided  into  parts;  and 
the  parts  are  named  according  to  their  number  and  size 

Divided  now  into  2  equal  parts,  ^m 
One  of  these  parts  is  one-half,  g 

1 .  How  many  halves  are  there  in  the  whole  of  any  thing? 

2.  Suppose  I  can  write  a  letter  on  1-haif  of  a  sheet  of 
paper;  how  much  paper  shall  I  use,  in  writing  2  letters  ^ 

3.  How  much  is  1-half  and  1-half,  added  together? 

Divided  now  into  8  equal  parts. 
One  of  these  parts  is  one-third.  \ 

4.  How  many  thirds  are  there  in  the  whole  of  any  thing? 

5.  If  a  carpenter  can  make  3  door-panels  of  1  board, 
what  part  of  one  board  will  he  use,  in  making  1  panel  ? 

6.  Which  is  the  greater  part,  1-half,  or  1 -third? 

Divided  now  into  4  equal  parts.  I 
One  of  these  parts  is  one-fourth, 

7.  How  many  fourths  are  there  in  the  whole  of  1  thing? 

8.  I  gave  1 -fourth  of  an  orange  to  John,  and  2-fourths 
to  Frances.     How  much  of  the  orange  did  I  give  a|ray  ? 

9.  Which  is  the  greater  part,  1 -third,  or  1 -fourth? 

Divided  now  into  5  equal  parts. 
One  of  these  parts  is  one-ffth. 

10.  How  many  fifths  are  there  in  the  whole  of  any  thing? 

11.  Charles  divided  a  melon,  equally  among  5  boys. 
What  part  of  the  melon,  [how  many  fifths,]  had  2  boys  t 

12.  Which  is  the  smaller  part,  1 -fourth,  or  1 -fifth  ? 


I  FRACTIONS.  4S 

Divided  now  into  6  equal  parts.  _^^ 
One  of  these  parts  is  one-sixth.   '^^ 

13.  Howmany  sixths  are  there  in  the  wholeof  any  thing.^ 

14.  If  3  girls  and  2  boys  should  each  of  them-eat  1-sixth 
of  a  pie,  what  part  of  the  whole  pie  would  they  all  eat  ? 

15.  Which  is  the  greater  part,  l-fifth,  or  1-sixth.^ 


Divided  now  into  7  equal  parts,  ^m^m 

16.  How  many  sevenths  are  there  in  the  whole  of  1  thing.'* 

17.  John  broke  ofF2-sevenths  of  a  new  pencil,  and  cut 
off  1-seventh  more.     How  much  of  it  was  then  wasted  ^ 

18.  Which  is  the  smaller  part,  1-sixth,  or  1-seventh? 

Divided  now  into  8  equal  parts.  ^s=^^^^^^^^^s^^^g 

19.  How  many  eighths  are  there  in  the  whole  of  1  thing .^ 

20.  If  a  boy  earn  3  eighths  of  one  dollar,  and  find  4- 
eighths  mora,  what  part  of  one  dollar  will  he  then  have  ^ 

21.  Which  is  the  smaller,  1-seventh,  or  1 -eighth .? 

Divided  now  into  9  equal  parts.  ^^^J^p^M^^^^^^s^P^^ 
One  of  these  parts  is  one-ninth.  ^^^g^B^p^j^s^g^s^^ 

22.  How  many  ninths  are  there  in  the  whole  of  1  thing.'* 

23.  Stephen  paid  3-ninths  of  all  his  money  for  a  slate, 
and  6-ninths  for  a  blank-book.     How  much  had  he  left.'* 

24.  Which  is  the  greater  part,  1-eighth,  or  1-ninth? 

Divided  now  into  10  equal  parts,  ^g^ 
One  of  these  parts  is  one-tenth,  ^g^ 

25.  Howmany  tenths  are  there  in  the  whole  of  1  thing.** 

26.  If  a  book  cost  5-tenths  of  a  dollar,  and  a  penknife 
cost  4-tenths,  what  part  of  1  dollar  will  they  both  cost  .'* 

27.  Which  is  the  greater  part,  1-ninth,  or  1-tenth  ? 

Remark  1st.  It  appears  from  the  examples  above,  that, 
ONE-HALF  of  any  thing,  is  one  of  two  equal  parts  of  the 
thing; — ONE-THIRD  of  any  thing,  is  one  of  three  equal 
parts  of  the  thing; — ONE-FOURTH  of  any  thing,  is  one  ot 
four  equal  parts  of  the  thing;  and  so  on. 

Remark  2nd.  The  greater  the  number  of  parts  is,  into 
which  any  thing  is  divided,  the  smaller  the  parts  are. 


46 


ORAL    ARITHMETIC. 

Section  2. 


VI 


Note  to  Teachers.  One  object  in  this  section  is,  to  lead  the  pupil  to  apply 
rxjrrectly  the  terms  expressing  fractional  parts.  Every  answer,  therefore,  must 
oe  given  in  a  vulgar  fraction,  unreduced.  For  example,  two  fourths  is  the 
answer  which  must  be  given  to  the  3d  question.  The  books  to  be  closed  dur 
ing  tlie  recitation  of  this  section. 

1.  If  we  divide  any  thing  into  2  equal  parts,  and  take 
away  1  of  the  parts,  how 


much  of  the  thing  is  left } 

2,  If  we  divide  any  thing  into  3  equal  parts,  and  take 
away  2  of  the  parts,  how 


much  of  the  thing  is  left  .'* 

3.  If  we  divide  any  thing  into  4  equal  parts,  and  take 
away  2  of  the  parts,  how 
much  of  the  thing  is  left  ? 

4.  If  we  divide  any  thing  into  5  equal  parts,  and  take 
away  3  of  the  parts,  how 
much  of  the  thing  is  left  ? 

5.  If  we  divide,  any  thing  into  6  equal  parts,  "and  take 
away  3  of  the  parts,  how 


much  of  the  thing  Is  left  ? 

6.  If  we  divide  any  thing  into  7  equal  parts,  and  take 
away  2  of  the  parts,  how 


much  of  the  thing  is  left  ^ 

7.  If  we  divide  any  thing  into  8  equal  parts,  and  take 
away  5  of  the  parts,  how 
much  of  the  thing  is  left  ? 

8.  If  we  divide  any  thing  into  9  equal  parts,  and  take 
away  3  of  the  parts,  how 


much  of  the  thing  is  left  ? 

9.  If  we  divide  any  thing  into  10  equal  parts,  and  take 
away  4  of  the  parts,  how 


much  of  the  thino;  is  left  ? 


10.  Into  how  many  parts  must  any  thing  be  divided^ 
so  that  1  part  shall  be  1-eleventh  ? — Into  how  many,  so 
that  1  part  shall  be  1 -twelfth  ? — 1 -thirteenth  ^ 


2    3.  FRACTIONS.  47 

Section  3. 

Note  to  Teachers.  The  learners  may  be  referred  to  Remark  Ist.  under 
Uie^first  section  of  examples  in  this  chapter,  for  a  correct  form  of  expression 
to  be  adopted  in  answering  the  1st.,  4th.,  7lh.,  and  other  similar  questions  .n 
this  section.     Books  to  be  closed  during  tiie  recitation  of  this  section. 

1.  What  is  meant  by  one-half  oi  any  thing  } 

2.  Suppose  you  have  1-half  of  1  dollar; — what  part 
of  a  dollar  more  must  you  get,  to  make  up  1  dollar  ? 

3.  How  many  halves  are  equal  to  a  whole  one  ^ 

4.  What  is  meant  by  one-third  of  any  thing  .'' 

5.  If  I  should  cut  1  orange  into  thirds,  and  give  you 
2-thirds  of  it,  what  part  of  an  orange  would  you  still  want, 
to  make  up  1  orange  by  joining  the  parts  together  ? 

6.  How  many  thirds  are  equal  to  a  whole  one  ? 

7.  What  is  meant  by  one-fourth  of  any  thing  ? 

8.  Suppose  you  have  1-fourth  of  1  dollar, — what  part 
of  a  dollar  must  you  get,  to  make  up  1  dollar  ? 

9.  How  many  fourths  are  equal  to  a  whole  one  ? 

10.  What  is  meant  by  one-fifth  of  any  thing  } 

11.  If  I  should  cut  1  apple  into  fifths,  and  give  you 
•  4-fifths  of  it,  what  part  of  an  apple  would  you  still  want, 

to  make  up  1  apple  by  joining  the  parts  together  } 

12.  How  many  fifths  are  equal  to  a  whole  one  } 

13.  What  is  meant  by  one-sixth  of  any  thing  ? 

14.  If  I  own  2-sixths  of  1  acre  of  land,  and  I  wish  to 
own  1  acre,  what  part  of  1  acre  must  I  buy } 

15.  How  many  sixths  are  equal  to  a  whole  one  .'^ 

16.  What  is  meant  by  one-seventh  of  any  thing  .^ 

17.  A  man  bought  4-sevenths  of  a  pound  of  tea  at  one 
shop,  and  enough  more  at  another  shop  to  make  1  pound. 
What  part  of  1  pound  did  he  buy  at  the  last  shop  ? 

18.  How  many  sevenths  are  equal  to  a  whole  one  } 

19.  What  is  meant  by  one-eighth  of  any  thing  .-^ 

20.  James  had  5-eighths  of  a  dollar  given  him,  and  he 
earned  3-eighths  more.    How  much  money  had  he  then  -^ 

21.  How  many  eighths  are  equal  to  a  whole  one? 

22.  What  is  meant  by  one-ninth  of  any  thing }' 

23.  If  I  have  7-ninths  of  1  acre  of  land,  and  I  wish  to 
own  1  acre,  what  part  of  1  acre  must  I  buy } 

24.  How  many  ninths  are  equal  to  a  whole  one  ? 


48  ORAL    ARITHMETIC.  VI. 

25.  What  is  meant  by  one-tenth  of  any  thing  ? 

26.  Suppose  you  have  8-tenths  of  1  dollar, — what  part 
of  a  dollar  must  you  get,  to  make  up  1  dollar  ? 

27.  How  many  tenths  are  equal  to  a  whole  one  ? 

RELATIONS    OF    NUMBERS. 

Section   4. 

Note  to  Teachers.  The  object  of  this  section  is,  to  show  the  correspondence 
of  the  division  of  a  unit,  with  the  division  of  a  collection  of  units.  The  ques- 
tions that  inquire,  what  part  of  one  number  is  another  number,  must  be  answered 
ii5  the  terms  of  vulgar  fractions.  For  instance, — the  answers  required  in  the 
9th  example  are, — 1  is  1-fourth  of  4;  2  is  2- fourths  of  4;  3  is  3-fourths  of  4. 

A  collection  of  units  is  now  to  be  viewed  as  a  single  thing;  therefore  the 
Verb  singular  Avill  be  used  thus — 3  times  4  is  12. 

1.  If  1-half  of  a  sheet  of  paper  be  worth  1  cent,  what 
is  a  whole  sheet  worth  ? 

2.  Suppose  2  cents  are  lying  upon  the  desk  before  us; 
— v/hat  part  of  the  2  cents  is  1  cent  ? 

3.  What  part  of  2  is  1  ? 

4.  If  1 -third  of  a  loaf  of  bread  be  worth  1  cent,  what 
is  2-thirds  of  it  worth  ^     What  is  a  whole  loaf  worth  ? 

5.  Suppose  3  cents  are  in  a  pile  before  us; — what  part 
of  the  pile  is  1  cent  ?     What  part  of  the  pile  is  2  cents  ? 

6.  What  part  of  3  is  1  ?     What  part  of  3  is  2  ? 

7.  If  1-fourth  of  a  yard  of  ribbon  cost  1  cent,  what  will 
2Tfourths  of  a  yard  cost  ?  What  will  3-fourths  of  a  yard 
tost }     What  will  a  whole  yard  cost  ? 

8.  Suppose  4  cents  are  in  a  pile  before  us; — what  part 
of  the  pile  is  1  cent  ?     is  2  cents?     is  3  cents  ? 

9.  What  part  of  4  is  1  :     is  2  ?     is  3  .^ 

10.  If  1 -fifth  of  a  barrel  of  flour  be  worth  1  dollar, 
what  is  2-fifths  of  a  barrel  worth  ?  3-fifths  of  a  barrel  ? 
4'"fifths  of  a  barrel  ?     What  is  1  barrel  worth  ? 

1 1 .  What  part  of  5  is  1  ?     is  2  ?     is  3  ?     is.  4  ? 

12.  If  1-sixth  of  a  yard  of  ribbon  cost  1  cent,  what  will 
2-sixths  of  a  yard  cost  ?  3-sixths  of  a  yard  ?  5-sixths 
of  a  yard  ?     What  will  1  yard  cost  ? 

13.  What  part  of  6  is  1  ?    is  2  ?    is  3  f    is  4?     is  5  ? 

14.  If  a  horse  trot  1  mile  in  1 -seventh  of  an  hour,  how 
many  miles  will  he  trot  in  2-sevenths  of  an  hour  ?  in 
6-serenths  of  an  hour  ?     How  many  miles  in  1  hour  r 

15.  What  part  of  7  is  1  ?     is  2  ?     is  3  .^     is  4  .?     is  6  .^ 


4.  5.  RELATIONS    OF    NUMBERS.  49 

16.  If  1 -eighth  of  a  bar  of  silver  be  worth  1  dollar, 
what  is  3-eighths  of  the  bar  worth  ?  What  is  5-eighths 
of  the  bar  worth  ?     What  is  the  whole  bar  worth  ? 

17.  What  part  of  8  is  1  ?     is  2  ?     is  3  ?     is  5  ?     is  7  ? 
IS.  If  1-ninth  of  a  pound  of  sugar  cost  1  cent,  what 

will  2-ninths  of  a  pound  cost  ?     What  will  8-ninths  of  a 
pound  ?     What  will  1  pound  cost  ? 

19.  What  part  of  9  is  1  ?     is  2  ?     is  4  ?     is  6  ?     is  8  ? 

^     20.   If  a  man  can  build  1  rod  of  fence  in  1-tenth  of  a 

day,  how  many  rods  can  he  build  in  4-tenths  of  a  day  ? 

in  (3-tenths  of  a  day  ?     How  many  rods  in  1  day  ? 

21.  What  part  of  10  is  1.?     is  2  ?     is  3  .^^     is  4.^     is  6  .^ 

Section  6. 

1.  A  carpenter  having  sawed  a  board  into  halves,  finds 
by  measuring,  that  1-haIf  of  the  board  is  2  feet  long. 
How  long  was  the  whole  board  ? 

2.  Suppose  2  is  1-half  of  some  number,- — what  is  the 
whole  of  the  number  ? 

3.  If  1-haIf  of  a  pound  of  rice  be  worth  3  cents,  what 
is  a  whole  pound  worth  ? 

4.  3  is  1-half  of  what  number  ?  4  is  1-half  of  what 
number?     7  is  1-half  of  what  number  ? 

5.  How  many  times  1-half  of  any  number  will  nfiake 
the  whole  number  ? 

6.  There  is  just  room  for  2  boys  to  sit  upon  1 -third 
of  a  certain  board.  What  number  of  boys  could  sit  up- 
on the  whole  of  that  board  ? 

7.  Suppose  2  is  1-third  of  some  number, — what  is  the 
whole  of  the  number  ? 

8.  If  1 -third  of  a  box  of  raisins  be  worth  3  dollars, 
what  is  the  whole  box  worth  ? 

9.  3  is  1-third  of  what  number  ?  4  is  1-third  of  what 
number  ?     G  is  1-third  of  what  number  ? 

10.  How  many  times  l-third  of  any  number  will  make 
the  whole  number  ? 

11.  If  1 -fourth  of  a  yard  of  broad-cloth  cost  2  dollars, 
what  will  a  yard  cost  ? 

12.  2  is  1-fourth  of  what  number  ? 

Solution,     2  is  1-fourth  of  4  times  2.     4  times  2  is  8 

E 


OO  ORAL    ARITHMETIC.  VL 

13.  If  a  ship  sail  3  miles  in  l-foiirth  of  an  hour,  how 
many  miles  will  she  sail  in  an  hour  ? 

14.  3  is  1 -fourth  of  what  number?  4  is  1-fourth  of 
what  number  ?     10  is  1-fourth  of  what  number  ? 

15.  How  many  times  1  -fourth  of  any  number  will  make 
the  whole  number  ? 

16.  3  men  reaped  1 -fifth  of  a  field  of  wheat  in  a  day. 
What  number  of  men  would  have  reaped  the  whole  field  ? 

Sohition.     If  3  men  reaped   l-fifth,  5  times  3  men' 
would  have  reaped  5-fifths,  or  the  whole. 

17.  If  l-fifth  of  a  pound  of  loaf  sugar  be  worth  4  cents, 
what  is  1  pound  worth  ? 

13.  4  is  l-fifdi  of  what  number  ?  5  is  l-fifth  of  what 
number  ?     8  is  l-fifth  of  what  number  ? 

19.  How  many  times  l-fifth  of  any  number  will  make 
the  whole  number .'' 

20.  4  sheets  of  paper  is  1-sixth  of  a  quire.  How 
many  sheets  are  there  in  a  quire  f 

.21.  4  is  1-sixth  of  what  number  ?     2  is  1-sixth  of  what 
number  ?     3  is  1-sixth  of  what  number  ? 

22.  How  many  times  1-sixtli  of  any  number  will  make 
the  whole  number  ? 

23.  If  1 -seventh  of  a  ton  of  hay  be  worth  2  dollars, 
what  IS  a  whole  ton  worth  .^ 

24.  2  is  1-seventh  of  what  number.^  3  is  1-seventh 
of  what  number  ?     9  is  1-seventh  of  what  number  ? 

25.  5  is  1-eighth  of  what  number  ?  3  is  1-ninth  of 
what  number  ?     6  is  1 -tenth  of  what  number  ? 

SECttON    6. 

1 .  If  a  yard  of  cloth  be  worth  2  dollars,  what  is  1-half 
of  a  yard  worth  ?     What  is  1-half  of  2  ? 

Observation.  Smce  you  know  that  l-half  of  2  is  1, 
you  will  perceive,  that  l-half  of  any  other  number  must 
be  as  many  times  1  as  there  are  ticos  in  the  number. 

2.  What  is  1-half  of  4  ?     of  12  ?     of  18  ?     . 

3.  If  a  bar  of  silver  weigh  3  pounds,  what  will  1 -third 
of  it  weigh  ?     What  is  1 -third  of  3  ? 

Obsertation.  Here  notice,  that  1 -third  of  any  nu mbei 
is  as  many  times  1,  as  there  are  threes  in  the  number. 


6.  RELATIONS    OF    NUMBERS.  51 

4.  If  a  peck  of  oats  cost  9  cents  what  will  1 -third  of 
a  peck  cost  ?     What  is  1 -third  of  9  ? 

5.  What  is  1 -third  of  6  ?     of  15  ?     of  24  ? 

6.  If  a  man  earn  4  shilhngs  in  1  day,  how  much  does 
he  earn  in  1-fourth  of  a  day  ?     What  is  1-fourth  of  4  ? 

Observation.     Here  notice  that  1-fourth  of  any  number 
is  as  many  times  1,  as  there  are  fours  in  the  number? 

7.  If  a  pound  of  raisins  cost  20  cents,  what  will  1-fourth 
of  a  pound  cost  ?     What  is  1 -fourth  of  20  ? 

8.  What  is  1-fourth  of  8  ?     of  28  ?     of  64  ? 

9.  If  a  yard  of  cambric  cost  5  dimes,  what  will  1 -fifth 
of  a  yard  cost  ?     What  is  1 -fifth  of  5  ? 

10.  If  3,5  drums  of  figs  w^ill  pay  for  a  hogshead  of  su- 
gar, how  many  drums  will  pay  for  1 -fifth  of  a  hogshead  ? 

1 1 .  What  is  1-fifth  of  35  ?     of  25  ?     of  40  ? 

12.  If  a  man  earn  6  dollars  in  a  week,  how  much  does 
he  earn  in  1 -sixth  of  a  week  ?     What  is  1-sixth  of  6  ? 

13.  Suppose  writing  paper  costs  24  cents  a  quire; — 
what  will  be  the  price  of  1-sixth  of  a  quire  ? 

Solution,     If  a  quire  cost  24  cents,  1-sixth  of  a  quire 
will  cost  1-sixth  of  24  cents.     1-sixth  of  24  is  4. 

14.  What  is  1-sixth  of  12  ?     of  36  .?     of  48  ? 

15.  What  number  of  days  is  1 -seventh  of  a  week  ? 

16.  If  you  eat  21  meals  in  a  week,  how  many  do  you 
eat  in  1 -seventh  of  a  week,  or  1  day  ? 

17.  What  is  1-seventh  of  14  ?     of  21  ?     of  5.6  .? 

IS.   If  a  melon  weigh  8  pounds,  what  is  the  weight  of 
1-eighth  of  it  ^     What  is  1-eighth  of  8  ? 

19.  If  a  yard  of  silver  wire  cost  32  cents,  what  will 
1-eighth  of  a  yard  cost } 

20.  What  is  1-eighth  of  16  .?     of  32  ?     of  64  } 

21.  If  a  drum  of  figs  weigh  9  pounds,  what  is  the 
weight  of  1-ninth  of  a  drum  ^     What  is  1-ninth  of  9  } 

22.  Suppose  a  watch-chain  consists  of  18  links; — how 
many  links  are  there  in  1-ninth  of  the  chain  ? 

23.  What  is  1-ninth  of  18  ?     of  36  ?     of  72  ? 

24.  W'hat  number  of  cents  is  1-tenth  of  a  dime  ? 

25.  If  a  chest  of  Souchong  tea  be  worth  30  dollars, 
what  is  1-tenth  of  a  chest  worth  ^ 

26.  What  is  1-tenth  of  30  }     of  50  }     of  100  ? 


53  ORAL    ARITHMETIC.  VI. 

Section   7. 

Note  to  Teachers.  In  solving  the  following  questions,  the  learner  should 
nrst  state  what  proportional  part  of  the  number  to  be  divided  will  be  the  answer, 
and  tlience  proceed  to  find  the  answer  in  tlie  denomination  of  tlie  dividend.— 
See  solution  under  example  2d. 

1 .  If  any  number  of  oranges  should  be  divided 'equally 
between  2  boys,  what  part  of  the  number  would  1  boy 
receive  ?  What  part  would  1  boy  receive,  if  the  oranges 
were  divided  among  3  boys  }  4  boys  }  5  boys  }  6  boys  } 
7boys.^     8boys.^     Sboys.^     lOboys  ? 

2.  5  sailors  received  40  dollars,  which  they  divided 
equally  among  them.     What  did  1  sailor  receive  ? 

Solution,  If  5  sailors  received  40  dollars,  1  sailor 
must  have  received  1 -fifth  of  40  dollars.  1 -fifth  of  40 
dollars  is  8  dollars  ? 

3.  2  fishermen  caught  24  fishes,  which  they  shared 
equally.     How  many  was  each  man's  share  ? 

4.  If  a  traveller  spend  23  dollars  in  travelling  a  week, 
how  much  does  he  spend  a  day  ^ 

5.  A  farmer  can  keep  9  cows  on  36  acres  of  land: — . 
how  many  acres  would  it  take  to  keep  1  cow  } 

6.  3  men  have  a  bill  of  30  dollars  to  pay: — how  much 
must  each  man  pay  ? 

7.  If  a  stage  run  42  miles  in  6  hours,  what  distance 
does  it  run  in  1  hour  ? 

8.  Suppose  8  dozen  of  biscuit  to  be  worth  72  cents; — 
what  is  the  value  of  1  dozen  ^ 

9.  A  tailor  made  10  cloaks  of  40  yards  of  cloth.  How 
many  yards  did  he  put  into  each  cloak  ? 

•  Section  8. 

Note  to  Teachers.  Require  the  learner,  as  in  tlie  last  section,  to  com- 
mence every  solution  by  stating  what  proportional  part  of  the  given  number  is 
to  be  taken  for  the  answer. 

1 .  If  a  man  can  travel  35  miles  in  a  day,  what  distance 
can  he  travel  in  1 -seventh  of  a  day  ?  What  distance  in 
4-sevenths  of  a  day  ^ 

Solution.  If  he  can  travel  35  miles  in  a  day,  he  can 
travel  1 -seventh  of  35  miles  in  1 -seventh  of  a  day;  1 -sev- 
enth of  35  miles  is  5  miles. He  can  travel  4  times  5 

miles  in  4-sevenths  of  a  day;  4  times  5  miles  are  20  miles 


7.   8  RELATIONS   OF   NUMBERS.  53 

2.  There  are  24  sheets  of  paper  in  a  quire.  How 
many  sheets  are  there  in  1 -eighth  of  a  quire  ?  How  many 
sheets  in  3-eighths  of  a  quire  ?    . 

3.  What  is  1 -eighth  of  24  ?    3-eighths  of  24  ? 
Solution,     1 -eighth  of  24  is  3;  3-eighths  is  3  times  3. 

4.  If  a  pound  of  coffee  be  worth  15  cents,  what  is 
1-third  of  a  pound  worth  ?     2-thirds  of  a  pound  ? 

5.  What  is  1-third  of  15  ?     2-thirds  of  15  ? 

.    6.  If  a  bushel  of  barley  cost  50  cents,  what  does  1-fifth 
of  a  bushel  cost  ?     4-fifths  of  a  bushel  ? 
7..  What  is  1-fiflh  of  50  ?     4-fifths  of  50  ? 

6.  If  a  yard  of  ribbon  cost  28  cents,  what  will  1-fourth 
of  a  yard  cost  ?     3-fourths  of  a  yard  ? 

9.  What  is  1-fourth  of  28  ?     3-fourths  of  28  ? 

10.  If  an  acre  of  land  will  produce  30  bushels  of  rye, 
how  much  will  2-sixths  of  an  acre  produce  ? 

Direction.    First  get  what  1-sixth  of  an  acre  will  yield. 

11.  What  is  5-sixths  of  30  ? 

Direction.    First  get  1-sixth  of  30;  thence  5-sixths. 

12.  If  a  stage  run  81  miles  in  a  day,  what  number  of 
miles  will  it  run  in  7-ninths  of  a  day? 

13.  What  is  7-ninths  of  81.? 

14.  There  are  100  cents  in  a  dollar.  What  number 
of  cents  are  there  in  8-tenths  of  a  dollar  ? 

15.  What  is  8-ienths  of  100? 

16.  Albert's  kite  line  was  32  yards  long,  and  he  cut 
off  2-eighths  of  it  for  a  fish  hne.  What  was  the  length 
of  the  fish  line  ? 

17.  Suppose  a  boy  having  42  quills,  should  give  away 
3-sevenths  of  them; — how  many  would  he  give  away? — - 
and  how  many  vi  juld  he  have  left  ? 

18.  A  man  having  40  dollars,  paid  away  3-eighths  of 
his  money  for  a  Ion  of  hay.  What  was  the  price  of  the 
hay? — and  how  many  dollars  had  he  left  ? 

19.  A  boy,  who  had  45  cents,  paid  away  3-fifths  of 
hU  money  for  a  (juire  of  paper.  What  was  the  price  of 
th«  paper  ? — and  how  many  cents  had  he  left  ? 

20.  If  75  men  can  build  a  mile  of  fence  in  a'day,  whai 
number  of  men  njust  be  employed,  to  build  2-thirds  of  a 
mile  in  the  same  time  ? 


54  ORAL    ARITHMETIC.  VI 

Section    9. 

1 .  If  48  dollars  should  be  divided  equally  among  8  men, 
what  part  of  the  money, — and  what  number  of  dollars, 
would  3  men  receive  ? 

Solution.  3  men  would  receive  3-eighths  of  the 
money.  1 -eighth  of  48  dollars  is  6  dollars;  3-eighths 
is  3  times  6  dollars,  or  18  dollars. 

2.  A  tierce,  holding  42  gallons  of  molasses,  has  been 
emptied  into  6  kegs,  of  equal  size.  What  part  of  the 
molasses, — and  what  number  of  gallons  in  5  kegs  ? 

3.  What  part  of  6  is  5  ?     What  is  5-sixths  of  42  ? 

4.  If  a  piece  of  broad-cloth  containing  30  yards  will 
make  ten  suits  of  clothes,  what  part  of  the  piece, — and 
what  number  of  yards,  will  make  6  suits  ? 

5.  What  part  of  10  is  6  ?     What  is  6-tenths  of  30  ? 

6.  5  girls  had  15  oranges,  which  they  shared  equally: 
2  of  the  girls  gave  their  shares  to  a  sick  woman.  What 
part  of  15  oranges  did  the  woman  receive? — and  what 
number  of  oranges  did  she  receive  ? 

7.  W^hat  part  of  5  is  2  ?     What  is  2-fifths  of  15  ? 

8.  7  men  owned  56  sheep  in  company,  and  3  of  the 
men  took  out  their  shares.  What  part  of  the  flock, — and 
what  number  of  sheep  did  they  take  out  ? 

9.  What  part  of  7  is  3  ?     What  is  3-sevenths  of  56  ? 

10.  If  4  men  eat  28  biscuit  in  a  day,  what  part  of  28 
biscuit, — and  what  number  of  biscuit  will  2  men  eat  ? 

1 1 .  What  part  of  4  is  2  ?     What  is  2-fourths  of  28  ? 

12.  3  brothers  owned  60  acres  of  land  together,  and 
the  2  younger  sold  their  shares  to  the  oldest.  What  part 
of  the  land, — and  how  many  acres  did  they  sell  ? 

13.  What  part  of  3  is  2  ?     What  is  2-thirds  of  60  .? 

Section  10. 

1.  If  3  men  can  fell  18  trees  in  a  day,  how  many  trees 
can  4  men  fell  in  the  same  time  ? 

Solution.  If  3  men  can  fell  18  trees  in  a  day,  1  man 
can  fell  1-third  of  18  trees,  or  6  trees;  4  men  can  fell  4 
times  6  ti*^es,  or  24  trees. 

2.  What  is  4  times  1-third  of  18  ? 

Solution.     1  third  of  IS  is  6:  then  4  times  6  is  24.    ' 


9.    10.    11.       RELATIONS   OF   NUMBERS.  55 

3.  If  5  men  will  cut  20  cords  of  wood  in  a  day,  how 
many  cords  will  3  men  cut  in  the  same  time  ? 

4.  What  is  3  times  1 -fifth  of  20  ? 

5.  If  4  barrels  of  flour  cost  24  dollars,  how  much  will 
7  barrels  cost,  at  the  same  price  per  barrel  ? 

6.  What  is  7  times  1-fourth  of  24  ? 

7.  If  2  boats  will  carry  16  passengers  across  the  river, 
how  many  passengers  will  5  boats  carry? 

8.  What  is  5  times  1-half  of  16  ? 

9.  Suppose  a  cooper  can  make  27  barrels  in  9  days; — 
how  many  barrels  can  he  make  in  5  days  ? 

10.  What  is  5  times  1 -ninth  of  27  ? 

1 1 .  Suppose  6  kegs  will  hold  36  gallons  of  molasses; — 
what  number  of  gallons  will  4  kegs  hold  ? 

12.  What  is  4  times  1 -sixth  of  36  ? 

13.  If  8  soldiers  eat  56  pounds  of  beef  in  a  week,  how 
many  pounds  will  9  soldiers  eat  in  a  week  ? 

14.  What  is  9  times  1-eighth  of  56  ? 

15.  If  a  workman  can  earn  49  dollars  in  7  weeks,  how 
many  dollars  can  he  earn  in  6  weeks  ? 

16.  What  is  6  times  1-seventh  of  49  ? 

17.  If  10  casks  of  claret  wine  cost  80  dollars,  what 
would  be  the  price  of  8  casks  of  the  same  wine  ? 

Section    11. 

1.  Suppose  there  are  10  links  in  2-thirds  of  a  watch 
chain; — how  many  links  are  there  in  1 -third  of  the  chain.'* 
How  many  links  in  the  whole  chain  ? 

Solution.  If  there  be  10  Imks  in  2-thirds  of  the  chain, 
there  is  1-half  of  10  links  in  1-third  of  it:    1-half  of  10 

is  5. If  5  links  be  1-third  of  the  chain,  there  are  3 

times  5  hnks  in  the  chain:  3  times  5  is  15. 

2.  10  is  2-thirds  of  what  number  ? 

3.  If  3-fourths  of  a  pound  of  honey  cost  1 5  cents,  what 
will  1-fourth  cost?     What  will  a  pound  cost? 

4.  15  is  3-fourths  of  what  number? 

Solution.  Since  15  is  3-fourths  of  the  required  num- 
ber, 1-third  of  15  must  be  1-fourth  of  the  number:  1-third 
of  15  is  5.  If  5  be  1-fourth  of  the  number,  4  times  5,  o) 
20,  is  the  whole  number. 


56  ORAL    ARITHMETIC.  VI. 

5.  If  2-fifths  of  a  bushel  of  oats  cost  18  cents,  what 
will  1 -fifth  cost  ?     What  will  a  bushel  cost  ? 

6.  IS  is  2-fifths  of  what  number  ? 

7.  If  4-sevenths  of  a  kite  line  be  36  yards  long,  how 
long  is  1-seventh  ?     How  long  is  the  whole  line  ? 

8.  36  is  4-sevenths  of  what  number  ? 

9.  If  3-sixths  of  a  chest  of  tea  cost  21  dollars,  what 
will  1 -sixth  cost  ?     What  will  a  whole  chest  cost  ? 

10.  21  is  3-sixths  of  what  number? 

11.  If  5-eighths  of  a  pipe  of  wine  be  worth  30  dollars, 
what  is  the  value  of  the  whole  pipe? 

Direction.     First  find  what  1-eighth  is  worth. 

12.  30  is  5-eighths  of  what  number? 

13.  If  a  man  can  earn  40  cents  by  working  4-sevenths 
of  a  day,  how  much  can  he  earn  by  working  a  whole  day? 

14.  40  is  4-sevenths  of  what  number? 

15.  If  7-ninths  of  a  hogshead  of  sugar  be  worth  49 
dollars,  what  is  the  whole  hogshead  worth? 

16.  49  is  7-ninths  of  what  number? 

17.  If  a  rail-road  car  run  24  miles  in  8-tenths  of  an 
hour,  what  distance  will  it  run  in  an  hour? 

18.  24  is  S-tenths  of  what  number? 

19.  Henry  is  10  years  old;  and  his  age  is  equal  to 
5-sixths  of  Andrew's  age.     How  old  is  Andrew? 

Suggestion.  You  may  perceive  that  1 -fifth  of  Henry's 
Hge  must  be  equal  to  1-sixth  of  Andrew's  age. 

20.  10  is  5-sixths  of  what  number  ? 

21.  If  21  workmen  will  perform  3-fifths  of  a  certain 
piece  of  work  in  a  week,  what  number  of  workmen 
would  it  take  to  perform  the  whole  work  in  a  week  ? 

22.  21  is  3-fifihs  of  what  number? 

23.  A  coach-man  purchased  a  horse,  and  after  paying 
5-eighths  of  the  price,  he  still  owed  30  dollars.  What 
was  the  price  of  the  horse  ? 

Solution.  If  he  paid  5-eighths  of  the  price,  the  30 
dollars,  which  he  still  owed,  was  3-eighths  of  the  price. 
30  dollars  being  3-eighths  of  the  price,  1-third  of  30  dol- 
lars, or  10  dollars,  is  1-eighth.  10  dollars  being  1-eightb 
of  the  price,  S  times  10  dollars  is  the  price. 

24.  30  is  3-eighths  of  what  number? 


12.     »  RELATIONS   OF  NUMBERS.  57 

25.  If  2-fifthj  be  taken  from  the  whole  of  a/iy  thing, 
how  many  fifths  are  there  left  ? 

26.  While  George  was  fishing,  a  pickerel  broke  off 
2-fifths  of  his  line;  he  then  had  12  feet  of  the  line  left. 
How  long  was  his  fine  at  first  ? 

27.  Suppose  a  laborer  can  earn  60  cents  a  day,  by 
working  5-sixths  of  the  time; — how  much  could  he  earn 
by  working  constantly  ? 

28.  After  3-sevenths  of  a  cask  of  wine  had  leaked  out, 
the  owner  drew  off  the  remainder,  and  found  there  were 
48  gallons.     How  many  gallons  had  he  lost  ? 

29.  A  farmer  improved  3-ninths  of  his  farm  in  tillage, 
appropriated  4-ninths  to  pasturage,  and  had  18  acres  of 
wood-land.     How  many  acres  had  he  in  all  ? 

30.  2-eighths  of  Edward's  books  are  bound  in  leather, 
3-eighths  of  them  in  marble  paper,  and  15  of  them  in 
blue  paper.     How  many  has  he  of  each  description  ? 

Section  12. 

Note  to  Teachers.  Tliis  section  embraces  all  the  operations  taught  in  the 
preceding  sections  of  this  chapter.  Prefixed  to  each  example,  is  the  number  of 
the  section  in  which  the  operation  involved  in  the  example  is  taught.  If  the 
pupil  fail  in  any  part  of  this  section,  he  should  be  put  back  to  the  section  whose 
number  is  prefixed  ta  the  example  in  which  he  fails. 

REVIEW. 

1 .  ( §  1 .)  James  found  4-eigbths  of  a  dollar,  and  earn- 
ed 5-eighths.     How  much  money  had  he  then  .'* 

2.  (§2.)  If  we  divide  anything  into  6  equal  parts, 
and  take  away  4  parts,  how  much  of  the  thing  is  left  ? 

3.  (§  3.)  If  you  have  7-ninths  of  1  dollar,  what  part 
of  a  dollar  must  you  get  to  make  up  1  dollar  ? 

4.  (§  4.)  If  a  man  can  walk  1  mile  in  1-fourth  of  axx 
hour,  how  many  miles  can  he  walk  in  1  hour  ? 

5.  (§4.)  What  part  of  6  is  5  .^  What  part  of  7  is  3  .? 
What  part  of  10  is  4  ?     What  part  of  18  is  7  ? 

6.  (§  5.)  If  1-fourth  of  an  acre  of  land  will  produce 
9  bushels  of  corn,  how  much  will  1  acre  produce  ? 

7  (§5.)  If  1-^eighth  of  a  barrel  of  oeef  be  allowed 
to  3  soldiers  for  a  week's  provision,  what  number  of 
soldiers  will  1  barrel  supply  for  a  week  ? 


58  ORAL    ARITHIVIETIC.  VI. 

8.  (§  6.)  Suppose  a  yard  of  gold  w.re  to  be  worth 
28  dollars; — what  is  1 -fourth  of  a  yard  worth? 

9.  (§  7.)  If  50  day's  work  is  to  be  done  by  5  men, 
how  many  day's  work  must  each  man  perform  ? 

10.  (§8.)  If  one  pound  of  Hyson  tea  be  worth  96 
cents,  what  is  T-eigbths  of  a  pound  worth  ? 

11.  (§9.)  A  company  of  14  men  gave  84  dollars  for 
a  boat.     What  part  of  84  dollars  did  5  men  pay  ? 

12.  (§  9.)  If  8  dollars  will  buy  72  pounds  of  brown 
sugar,  how  many  pounds  will  6  dollars  buy.'^ 

13.  (§  10.)  Suppose  18  yards  of  cloth  will  make  6 
coats; — how  many  yards  are  required  for  10  coats  ^ 

14.  (§11.)  If  5-ninths  of  a  yard  of  cotton  cambric  be 
worth  50  cents,  what  is  the  value  of  one  yard  ^ 

15.  (§  11.)  Suppose  a  man  by  working  constantly 
can  dig  40  bushels  of  potatoes  in  a  day, — how  many 
bushels  will  he  dig,  if  he  be  idle  2-fifths  of  the  day  ? 

FRACTIONS  AND  RELATIONS. 

Section   13. 

Note  to  Teachers.  The  remainders^  that  will  arise  in  the  several  examples 
of  division  in  this  section,  must  be  expressed  in  the  language  of  fractions.  See 
answers  under  examj)le3  4th.  and  8th.  If  the  learner  should  not  readily  un- 
derstand the  process  of  converting  the  remainders  into  fractions,  he  may  be 
referred  to  section  4th.  in  tiiis  chapter,  and,  after  reviewing  the  examples 
therein,  may  return  immediately  to  this  section. 

1.  How  much  cloth,  at  2  dollars  a  yard,  can  I  buy  for 
1  dollar  }     How  much  for  3  dollars  } 

2.  What  part  of  2  is  1  }     How  many  times  2  in  3  .'' 

3.  How  many  yards  of  ribbon,  at  2  cents  a  yard,  can 
you  buy  for  7  cents  h — I  mean, —  how  many  whole  yards, 
and  what  part  of  another  yard  can  you  buy } 

4.  How  many  times  2  in  7  .^— I  mean, —  how  many 
twos  are  there,  and  what  part  of  1  niore  two^  in  7  } 

Solution.  2  is  contained  in 7, 3  times,  and  1  over:  the 
1  over  is  1-half  of  another  time  2.     Jlns,  3  and  1-half. 

5.  How  many  tunes  2  in  9  .^     in  12  ?     in  13  ? 

6.  How  much  wine,  at  3  dollars  a  gallon,  can  I  buf 
for  1  dollar  ?     How  much  for  4  dollars  ? 

7.  What  part  of  3  is  I  ?     How  many  times  3  in  4  ^ 


13.  FRACTIONS  AND  RELATIONS.  S9 

8.  If  a  girl  by  setting  types  can  earn  1  dollar  in  3 
days,  how  much  can  she  earn  in  20  days  ? 

Solution,  She  can  earn  as  many  dollars  as  there  are 
threes  in  20.  3  in  20,  6  times  and  2  over;  the  2  over  is 
2-lhirds  of  another  three.     Ans,  6  dollars  and  2-thirds. 

9.  How  many  times  3  in  4  ?     in  16  ?     in  20  ? 

10.  If  a  pound  of  lead  cost  4  cents,  how  much  can  I 
buy  for  1  cent  ?     How  many  pounds  for  9  cents  ? 

11.  If  it  take  a  man  1  hour  to  walk  4  miles,  how  many 
hours  will  it  take  him  to  walk  15  miles  ? 

12.  How  many  times  4  in  9  ?     in  15  .'^     in  34  ? 

13.  How  much  coal,  at  5  dollars  a  ton,  can  be  bought 
for  1  dollar  ?     How  much  for  7  dollars  ^ 

14.  How  many  hogsheads  of  salt,  at  5  dollars  a  hogs- 
head, can  be  bought  for  44  dollars  ^ 

15.  How  many  times  5  in  7  .''     in  44  }     in  58  } 

16.  How  much  ribbon,  at  6  cents  a  yard,  can  you  buy 
for  10  cents  ?     How  much  for  37  cents  "^ 

17.  If  it  cost  6  cents  a  mile  to  ride  in  the  stage,  what 
number  of  miles  can  you  ride  for  50  cents  ? 

18.  How  many  times  6  in  10  ?     in  50  ?     in  45  ^ 

19.  How  much  sugar,  at  7  cents  a  pound,  can  I  buy 
for  9  cents  ?     How  much  for  52  cents  ? 

20.  How  many  pounds  of  shot,  at  the  rate  of  7  cents  a 
pound,  must  be  sold  for  34  cents  ? 

21.  How  many  times  7  in  9  ?     in  52  ?     in  34  } 

22.  How  much  hay,  at  8  dollars  a  ton  must  be  sold  for 
9  dollars  ?     How  much  for  29  dollars  ? 

23.  How  many  pounds  of  honey,  at  the  rate  of  8  cents 
a  pound,  must  be  sold  for  77  cents  ? 

24.  How  many  times  8  in  9  ?     in  29  ?     in  77  ? 

25.  At  9  cents  a  pound,  how  much  cheese  must  be 
sold  for  13  cents  ?     How  much  for  31  cents  ? 

26.  If  a  man  work  for  9  cents  an  hour,  how  many 
hours  will  it  take  him  to  earn  64  cents  ? 

27.  How  many  times  9  in  13  ?     in  31  ?  .  in  64  }  , 

28.  How  much  sugar,  at  10  cents  a  pound,  can  be 
bought  for  12  cents  ?     How  much  for  64  cents  ? 

29.  If  a  workman  can  build  10  rods  of  fence  in  1  day, 
how  many  days  will  It  take  him  to  build  48  rods  } 


60  ORAL    ARITHMETIC.  VI, 

30.  How  many  times  10  in  12  ?     in  64  ?     in  48  ? 

31.  How  much  rice  at  3  cents  a  pound,  must  be  giveu 
for  4  quarts  of  milk  at  5  cents  a  quart  ? 

Direction.     First  find  the  vahie  of  4  quarts  of  milk. 

32.  How  many  times  3  in  4  times  5  ? 
Direction.     First  find  how  much  4  times  5  is. 

33.  How  many  pounds  of  flour  at  4  cents  a  pound, 
must  be  given  for  5  pounds  of  honey  at  7  cents  a  pound  ? 

34.  How  many  pmes  4  in  5  times  7  ? 

35.  What  quantity  of  butter  at  10  cents  a  pound,  wil. 
pay  for  6  combs  at  8  cents  apiece  ? 

36.  If  an  active  man  earn  7  shilhngs  a  day,  and  a  lazy 
man  4  shillings,  how  many  days  must  the  lazy  man  work, 
to  pay  the  active  man  for  working  6  days  ? 

37.  How  many  times  4  in  6  times  7  ? 

3S.  How  many  yards  of  cloth  at  5  dollars  a  yard,  will 
pay  for  3  boxes  of  raisins  at  9  dollars  a  box  ? 

39.  Hov\^  many  times  5  in  3  times  9  ? 

40.  What  quantity  of  corn  at  6  dimes  a  bushel,  will 
pay  for  1 1  bushels  of  oats  at  3  dimes  a  bushel  ? 

41.  How  many  times  6  in  11  times  3  ? 

Section   14. 

jVoie  to  Teachers.  The  lerirners  must  now  be  led  to  observe,  that  the  ex 
pressions,  l-/m//,  1-third,  2-iliirdSy  Sec.  are  to  be  understood, — l-haW  of  one 
1-third  of  one,  2-tliirds  of  one.  Sec.  in  all  cases  where  the  number,  cf  whicii 
the  fraction  indicates  a  part,  is  not  stated. 

1 .  If  I  should  cut  each  of  3  sheets  of  paper  into  halves, 
how  many  halves  would  they  make  ? 

Solution.  In  1  sheet  there  are  2-halves, —  in  3  sheets 
there  are  3  times  2-halves,  or  6-halves. 

2.  How  many  halves  are  there  in  1?     in  2  ?     in  3? 

3.  If  I  had  4  sheets  and  1-half  of  a  sheet  of  paper,  how 
many  boys  could  I  supply  with  half  a  sheet  apiece  ? 

4.  How  many  halves  are  there  in  4  and  1-half? 

5.  If  2  slate  pencils  should  each  of  them  be  broken 
into  thirds,  how  many  thirds  would  they  make  ? 

6.  How  many  thirds  are  there  in  1  .^     in  2  ?     in  4  ? 
7*.   Suppose  I  had  3  pencils  and  2-thirds,  how  many 

boys  could  I  supply  with  1-third  of  a  pencil  each  ^ 
8.  How  many  thirds  are  there  in  3  and  2'thirds  ? 


14.  FRACTIONS    AND    RELATIONS.  61 

9.  If  1 -fourth  of  a  yard  of  cloth  will  make  a  satchel, 
how  many  satchels  will  2  yards  make  ? 

10.  How  many  for.rths  are  there  in  1?     in  2  ?     in  3  } 

11.  How  many  quires  of  paper,  at  1 -fourth  of  a  dollar 
a  quire,  can  you  buy  for  3  dollars  and  2-fouiths  ? 

Solution.  I  can  buy  as  many  quires  as  there  are  fourths 
of  a  dollar,  in  3  dollars  and  2-fourths.  In  1  dollar  there 
are  4-fourths, —  in  3  dollars,  3  times  4-fourths,  or  12- 
fourths:   12-fourths  plus  2-fourths  is  14-fourths. 

12.  If  a  carpenter  use  1 -fifth  of  a  board  to  make  1  book 
shelf,  how  many  book  shelves  can  he  make  of  a  whole 
board  }  of  2  whole  boards  ?  of  2  boards  and  3-fiftl^s  ? 
of  4  boards  and  1 -fifth  ? 

13.  How  many  fifths  are  there  in  1  ?  in  2  ?  in  2  and 
3-fifths  ?     In  4  and  1-fifth  ? 

14.  If  a  bunch  of  quills  cost  1 -sixth  of  a  dollar,  how 
many  bunches  can  you  buy  for  1  dollar  ?  for  1  dollar 
and  5-sixths  ?     for  3  dollars  and  2-sixths  ? 

15.  How  many  sixths  are  there  in  1?  in  1  and  5- 
sixths  ?     in  3  and  2-sixths  ? 

16.  If  a  stagp  run  1  mile  in  1-seventh  of  an  hour,  what 
number  of  miles  will  it  run  in  1  hour  ?  in  2  hours  and 
1-seventh  ?     in  4  hours  and  4-sevenths  ? 

17.  How  many  sevenths  are  there  in  1  ?  in  2  and 
1-seventh  ?     in  4  and  4-sevenths  ? 

18.  At  1 -eighth  of  a  dollar  a  yard,  how  many  yards 
of  ribbon  can  I  buy  for  1  dollar  and  3-eighths  r  for  2 
dollars  ?     for  5  dollars  and  7-eighths  ? 

19.  How  many  eighths  are  there  in  1  and  3-eighths  .'* 
in  2?     in  5  and  7-eighths  ? 

20.  How  many  ninths  are  there  in  1.^  in  1  and  4- 
nmths  ?     in  2  ?     in  2  and  7-mnths  ?     in  6  ? 

21.  How  many  tenths  are  there  in  1.^  in  4.^  in  3 
and  5-tenths  ?     in  5  ?     in  8  and  3-tenths  ? 

22.  A  laborer  earned  9  dollars  and  a  half,  working  at 
half  a  dollar  a  day.      How  many  days  did  he  work  ? 

23.  If  1-eighth  of  a  yard  of  cloth  cost  a  dollar,  what 
will  3  yards  and  G-eii^hths  cost  .^ 

24.  If  a  man  earn  a  dollar  in  1 -sixth  of  a  week,  how 
much  can  he  earn  m  8  we3ks  and  4-sixths  ? 


02  ORAL    ARITHMETIC.  VI 

25.  If  it  take  1-fifth  of  a  pound  of  fur  to  make  a  hat, 
how  many  hats  can  be  made  of  4  pounds  and  2-fifths  ? 

26.  If  1-fourth  of  a  yard  of  late  cost  a  dollar,  how 
much  will  5  yards  and  3-fourths  of  a  yard  cost  ? 

Section   15. 

1.  How  many  dollars  in  2-halves  of  a  dollar?  in  3- 
halves  of  a  dollar?     Jlns.   1  dollar.     1  dollar  and  1-half. 

2.  How  many  dollars  are  there  in  4  half  dollars  ?  in 
5  half  dollars  ?     in  9  half  dollars  ? 

3.  How  many  whole  ones  in  2-halves  ?  in  3-halves  ? 
in  4-halves  ?     in  5-halves  ?     in  9-halves  ? 

4.  What  will  13  pencils  cost,  at  half  a  cent  apiece  ? 

5.  If  3-thirds  of  an  orange  be  put  together  they  make 
up  1  orange.  Now,  if  you  had  6-thirds  of  an  orange, 
how  many  oranges  could  you  make  up  ?  if  you  had 
10-thirds  ?     if  you  had  17-thirds  ? 

6.  How  many  whole  ones  in  6-thirds  ?  in  10-thirds  ? 
in  17-thirds  ? 

7.  What  cost  26  quills,  at  1-third  of  a  cent  apiece  ? 
Solution.     If  1  quill  cost  1-third  of  a  cent,  26  quills 

will  cost  26-thirds  of  a  cent.  26-thirds  of  a  cent  are  as 
many  cents  as  3  is  contained  times  in  26.  3  in  26,  S 
times  and  2  over.     *Rns.   S  cents  and  2-thirds. 

8.  How  many  whole  apples  could  you  make  up,  if 
you  had  5-fourths  of  an  apple  ?     14-fourths  of  an  apple  ? 

9.  What  cost  31  cups,  at  1-fourth  of  a  dollar  apiece  ? 

10.  How  many  whole  ones  in  5-fourths  ?  in  14- 
fourths  ?     in  31 -fourths  ? 

11.  If  1  cotton  ball  be  given  for  1-fifth  of  a  yard  of 
galloon,  how  much  galloon  must  be  given  for  8  cotton 
balls  ?     for  17  cotton  balls  ?     for  44  cotton  balls  ? 

12.  How  manv  whole  ones  are  there  in  8-fifths  ?  in 
l7-fifths  ?     in  44.fifths  ? 

13.  If  a  quire  of  paper  cost  1-sixth  of  a  dollar,  what 
IS  the  cost  of  12  quires  ?     17  quires  ?     19  quires  ? 

14.  How  many  whole  ones  in  12-sixths  ?  in  17- 
sixths  ?     in  19-sixths  ? 

15.  How  many  whole  ones  in  18-sevenths?  m  24- 
tenths  ?     in  31-eighths  ?     in  47-ninths  ?     in  25-fourtljs  ^ 


16.  17.    FRACTIONS  AND  RELATIONS.        63 

Section    16. 

1.  Ellen  paid,  for  the  Young  Ladies^  Class  Book^  3- 
fourths  of  a  dollar;  for  the  Boston  School  Atlas^  2-fourth3 
of  a  dollar;  and  for  the  J^ational  Spelling- Book^  1 -fourth 
of  a  dollar.     What  did  the  whole  cost  ? 

2.  How  much  is  3-fourths  and  2-fourths  and  1 -fourth  ? 

3.  A  trader  sold  a  piece  of  cloth  for  19  dollars  and 
5-eighths,  and  a  hat  for  4  dollars  and  7-eighths.  How 
many  dollars  did  he  receive  for  both } 

Solution,  19  dols.  plus  4  dols.  are  23  dols.  5- 
eighths  of  a  dol.  plus  7-eighths  of  a  dol.  are  12-eighths 
of  a  dol.,  equal  to  1  dol.  and  4-eighths.  Then,  23  dols. 
plus  1  dol.  and  4-eighths  are  24  dols.  and  4-eighths. 

4.  A  traveller  rode  31  miles  and  3-fifths  in  the  fore- 
noon, and  25  miles  and  4-fifths  in  the  afternoon.  How 
many  miles  did  he  ride  in  the  whole  day  ? 

5.  What  is  31  and  3-fifths  plus  25  and  4-fifths  ? 

6.  A  trader  bought  some  goods  for  64  dollars  and  5- 
sevenths,  and  paid  5  dollars  and  3-sevenths  for  the  post- 
age of  them.     What  was  the  whole  expense  ? 

7.  What  is  64  and  5-sevenths  plus  5  and  3-sevenths  ? 

8.  A  gentleman  paid  33  dollars  and  7-tenths  for 
some  cloth,  and  11  dollars  and  6-tenths  for  having  it 
made  into  a  suit  of  clothes.     What  did  the  suit  cost  ? 

9.  What  is  33  and  7-tenths  plus  11  and  6-tenths  ? 

10.  What  is  16  and  6-ninths  plus  8  and  5-ninths.^ 

11.  What  is  40  and  5-sixths  plus  41  and  3-sixths  ? 

Section    17. 

1.  Suppose  a  rail-road  car  to  run  2-thirds  of  a  mile 
in  1  minute,  what  distance  will  it  run  in  10  minutes  ? 

Solution.  In  10  minutes  it  will  run  10  times  2-thirds 
of  a  mile,  or  20-thirds  of  a  mile.  20-thirds  of  a  mile  are 
equal  to  6  miles  and  2-thirds. 

2.  If  3-fourths  of  a  gallon  of  wine  leak  out  of  a  cask 
in  1  hour,  how  much  will  leak  out  in  7  hours  ? 

3.  How  many  whole  ones  in  7  times  3-fourths } 

4.  If  a  yard  of  cambric  muslin  cost  4-fifths  of  a  dol- 
lar, how  much  will  9  yards  cost  ? 

5.  How  many  whole  ones  in  9  times  4-fifths  ^ 


ft4  ORAL    ARITHMETIC.  VI 

6.  Suppose  a  man  to  eat  5-sixths  of  a  pound  of  beef 
m  one  day,  how  many  pounds  will  he  eat  in  5  days  ? 

7.  How  many  whole  ones  in  5  times  5-sixths  ? 

8.  If  3-sevenths  of  a  pound  of  gunpowder  tea  cost  1 
dollar,  how  many  pounds  can  I  buy  for  8  dollars  ? 

9.  How  many  whole  ones  in  8  times  3-sevenths  ? 

10.  Suppose  5-eighths  of  a  yard  of  cloth  will  make  a 
vest,  how  many  yards  will  it  take  to  make  6  vests  ? 

11. .  How  many  whole  ones  in  6  times  5-eighths  ? 

12.  If  1  quire  of  letter  paper  be  worth  4-ninths  of  a 
dollar,  how  many  dollars  are  7  quires  worth  ? 

13.  How  many  whole  ones  in  7  times  4-ninths  ? 

14.  Suppose  a  man  to  walk  1  mile  in  2-tenths  of  an 
hour,  what  time  will  it  take  him  to  walk  9  miles  ? 

15.  How  many  whole  ones  in  9  times  2-tenths  ^ 

Section    18. 

1.  What  will  6  yards  of  broad-cloth  cost,  at  7  dollars 
and  3-eighths  of  a  dollar  per  yard  ? 

Solution.  6  yards  will  cost  6  times  7  dollars  and  3- 
eighths.  6  times  7  dollars  are  42  dollars.  6  times  3- 
eighths  are  18-eighths,  equal  to  2  and  2-eighths.  Then, 
42  dollars  plus  2  dollars  and  2-eighths  are  44  dollars 
and  2-eighths. 

2.  What  will  4  hundred- weight  of  sugar  cost,  at  9 
dollars  and  2-fifths  per  hundred-weight  ? 

3.  What  is  4  times  9  and  2-fifi;hs  ? 

4.  Suppose  a  ship  to  sail  10  miles  and  1-half  in  one 
hour,  what  distance  will  it  sail  in  7  hours  ^ 

5.  What  is  7  times  10  and  1-half? 

6.  If  a  horse  eat  1  bushel  and  9-tenths  of  a  bushel  ot 
oats  in  a  week,  how  much  will  he  eat  in  4  weeks  ? 

7.  What  is  4  times  1  and  9-tenths  ? 

8.  If  1  dime  will  buy  3  yards  and  2-thirds  of  a  yard 
of  ribbon,  how  many  yards  will  6  dimes  buy  ? 

9.  What  is  6  times  3  and  2-thirds  ? 

10.  Suppose  the  price  of  coal  at  the  mine,  is  3  dol- 
lars a  ton,  and  the  freight  of  it  to  the  city  is  3-fourths 
of  a  dollar  a  ton,  w^hat  will  10  tons  cost  at  the  city  } 

11.  What  is  10  times  3  and  3-fourths  ? 


18.  19.    FRACTIONS  AND  RELATIONS.        65 

12.  Suppose  a  boat  goes  10  miles  and  6-sixths  of  a 
mile  in  1  hour^  what  distance  will  it  go  in  8  hours  ? 

13.  What  is  8  times  10  and  5-sixths  ? 

14.  If  3  yards  and  7-eighths  of  cloth  will  make  a 
cloak,  how  many  yards  will  it  take  to  make  5  cloaks  } 

15.  What  is  5  limes  3  and  7-eighths  ? 

Section    19. 

1.  If  1-fifth  of  a  chest  of  tea  be  worth  6  dollars  and 
7-eighths,  what  is  a  whole  chest  worth  ? 

Solution,  A  whole  chest  is  worth  5  times  as  much 
as  1-fifth  of  a  chest.  5  times  6  dollars  are  30  dollars; 
5  times  7-eighths  of  a  dollar  are  35-eighths  of  a  dollar, 
or  4  dollars  and  3-eighths.  30  dollars  plus  4  dollars 
and  3-eighths  are  34  dollars  and  3-eighths. 

2.  6  and  7-eighths  is  1-fifth  of  what  number  ? 

3.  Suppose  1 -ninth  of  a  kite  line  to  be  5  yards  and 
3-fourths  of  a  yard  long, — how  long  is  the  whole  line  ? 

4.  5  and  3-fourths  is  1 -ninth  of  what  number  ? 

5.  A  young  man  being  asked  his  age,  answered  indi- 
rectly, that  1 -third  of  his  age  was  7  years  and  2-sixths 
of  a  year.     What  was  his  age  ? 

6.  7  and  2-sixths  is  1 -third  of  what  number  ? 

7.  Suppose  a  man  can  build  3  rods  and  2-fifths  of  a 
rod  of  wall  in  1-sixth  of  a  week, — how  many  rods  can 
he  build  in  a  whole  week  ? 

8.  3  and  2-fifths  is  1-sixth  of  what  number? 

9.  If  1 -tenth  of  a  bushel  of  corn  be  worth  6  cents  and 
I -fourth  of  a  cent,  what  is  a  bushel  worth  ? 

10.  6  and  1 -fourth  is  1 -tenth  of  what  number? 

11.  1-fourth  of  Edmund's  kite  line  measures  8  yard« 
and  3-sevenths  of  a  yard.     How  long  is  the  line  ? 

12.  8  and  3-sevenths  is  1-fourth  of  what  number  ? 

13.  If  1-half  of  a  yard  of  lace  cost  3  dollars  and  4- 
fifths  of  a  dollar,  what  will  a  yard  cost  ? 

14.  3  and  4-fifths  is  1-half  of  what  number? 

15.  Suppose  that  1-seventh  of  an  acre  of  land  will 
produce  6  bushels  and  7-ninths  of  a  bushel  of  barley 
how  many  bushels  will  an  acre  produce  ? 

16.  6  and  7-ninths  is  1-seventh  of  what  number  ? 


ORAL    ARITHMETIC 

Section  20. 


VI. 


1.  If  1  apple  were  divided  equally  among  3  boys,  what 
part  of  1  apple  would  1  boy  receive  ?  If  2  apples  were 
thus  divided,  how  many  thirds  would  one  boy  receive  ? 

2.  Here  we  see  1 -third 
of  2  boards,  placed  over  2- 
thirds  of  1  board.  Is  it  not 
plain,  that  1 -third  of  the  2 
boards  together,  is  equal  to 
2-thirds  of  1  board  ? 

3.  1-third  of  2  is  equal  to  what  part  of  1  ? 

4.  There  were  3  boys,  who  had  1  dollar  apiece;  and 
each  boy  gave  a  decrepit  soldier  1-fourth  of  his  money. 
What  part  of  1  dollar  did  the  poor  soldier  receive  ? 

5.  Here,  we  see  lifourth 
of  3  boards  placed  over  3- 
fourths  of  1  board.  Sup- 
pose the  fourths  seen  in  the 
3  boards  should  be  placed 
together  end  to  end — Is  it 
not  plain,  they  would  make 
3-fourths  of  1  board  ? 

6.  1-fourth  of  3  is  equal  to  what  part  of  1  ? 

,  7.  I  have  4  oranges  to  divide  among  5  boys. — I  first 
cut  1  orange  into  fifths,  and  give  each  boy  1-fifth;  and 
thus  I  proceed,  dividing  1  orange  at  a  time,  until  they 
are  all  divided.  Now,  what  part  of  a  whole  orange  can 
each  boy  make  up,  by  joining  his  fifths  together  ? 

8.  1-fifth  of  4  is  equal  to  what  part  of  1  ? 

9.  If  1  melon  were  divided  equally  among  6  boys, 
what  part  of  1  melon  would  1  boy  receive  ?  If  2  melons 
were  divided,  how  many  sixths  would  1  boy  receive  } 

10.  1 -sixth  of  2  is  equal  to  what  part  of  1  ? 

Jl.  If  3  barrels  of  fiour  were  divided  equally  among 

7  men,  how  much  would  1  man  receive  ? 

12.  1-seventh  of  3  is  equal  to  what  part  of  1  ? 

13.  If  3  pounds  of  beef  were  divided  equally  among 

8  soldiers,  what  part  of  a  pound  would  1  soldier  get  ? 

14.  1 -eighth  of  5  is  equal  to  what  part  of  1  ? 


20.   21.        FRACTIONS   AND   RELATIONS.  67 

f 

15.  An  ostler  has  2  bushels  of  oats  to  divide  among 
9  horses; — how  much  must  he  give  to  each  horse  ^ 

16.  1-nmth  of  2  is  equal  to  what  part  of  1? 

17.  If  7  dollars  were  divided  equally  among  10  men, 
what  part  of  1  dollar  would  each  man  have  ? 

18.  1-tenth  of  7  is  equal  to  what  part  of  1? 

19.  1-fourth  of  2  is  equal  to  what  part  of  1  ? 

20.  1-sixth  of  5  is  equal  to  what  part  of  1.'^ 

21 .  1 -eighth  of  3  is  equal  to  what  part  of  1  ? 

22.  There  were  36  oranges  in  a  basket  and  Albert 
was  directed  to  take  1-fourth  of  them.  Accordingly  he 
cirt  1-fourth  out  of  every  orange,  and  took  it  to  himself. 
How  many  fourths  of  an  orange  did  he  get  ?  He  then 
joined  his  fourths  together,  to  make  them  into  whole 
oranges; — how  many  whole  ones  had  he? 

23.  1-fourth  of  36  is  equal  to  how  many  fourths  of  1  ? 
— equal  to  how  many  whole  ones  ? 

24.  In  another  basket  there  were  also  36  oranges, 
and  Benjamin  was  directed  to  take  1-fourth  of  them. 
But,  instead  of  cutting  1-fourth  out  of  every  orange,  as 
Albert  did,  he  took  1  orange  from  every  4  in  the  basket. 
How  many  oranges  did  Benjamin  get? 

25.  Now  tell  me  which  is  the  most; — 1-fourth  of  36, 
or  36-fourths  of  1  ? 

26.  1-half  of  10  dollars  is  equal  to  how  many  halves 
of  1  dollar? — equal  to  how  many  dollars? 

27.  1 -third  of  18  oranges  is  equal  to  how  many  thirds 
of  1  orange? — equal  to  how  many  whole  oranges? 

28.  1 -fifth  of  17  oranges  is  equal  to  how  many  fifths 
of  1  orange?— equal  to  how  many  whole  oranges? 

29.  1-sixth  of  42  is  equal  to  how  many  sixths  of  1? 
— equal  to  how  many  whole  ones? 

30.  1 -seventh  of  56  is  equal  to  how  many  sevenths 
of  1? — equal  to  how  many  whole  ones? 

Section   21. 

1.  If  a  chest  of  green  tea  be  worth  27  dollars,  what 
is  1-fourth  of  it  worth? 

Solution.  1-fourth  of  the  tea  is  worth  1-fourth  of  27 
dollars.     1-fourth  of  27  dollars  is  6  dollars,  there  being 


68  ORAL    ARlTHxMETIC.  VI 

3  dollars  over.     1 -fourth  of  3  dollars  is  equal  to  3-fourths 
of  1  dollar.     6  dollars  plus  3-fourth3  of  a  dollar  are  6 

dollars  and  3-fourths Or,  we  may  say, — One  fourth 

of  27  dollars  is  27-fourths  of  1  dollar;  equal  to  6  dollars 
and  3-fourths. 

2.  What  is  1 -fourth  of  27? 

3.  3  men  bought  a  barrel  of  sugar  for  23  dollars,  and 
divided  it  equally  among  them,  each  man  taking  1-third 
of  the  sugar,  and  paying  1-third  of  the  price.  How 
many  dollars  did  each  man  pay? 

4.  What  is  1-third  of  23? 

5.  Suppose  a  family  to  eat  26  loaves  of  bread  m  a 
week; — what  number  of  loaves  would  the  family  con- 
sume in  1 -seventh  of  a  week,  or  1  day? 

6.  what  is  1 -seventh  of  26  ? 

7.  Suppose  48  bushels  of  wheat  are  to  be  divided 
among  5  men;  how  much  will  1  man  receive  ? 

8.  What  is  1 -fifth  of  48? 

9.  6  men  purchased  a  boat  for  27  dollars:  each  man 
paid  1 -sixth  of  the  money,  and  owned  1 -sixth  of  the 
boat.     How  many  dollars  did  1  man  pay? 

10.  What  is  1-sixth  of  27? 

11.  Suppose  a  bag  of  coffee  to  weigh  65  pounds; — 
what  is  the  weight  of  1 -ninth  of  it  ? 

12.  What  is  1-ninth  of  65? 

13.  If  it  will  take  a  man  60  days  to  clear  a  piece  of 
wood-land,  in  what  time  will  he  clear  1-eighth  of  it? 

14.  What  is  1-eighth  of  60? 

15.  A  sailor  was  cast  upon  a  desolate  island,  and 
subsisted  10  days  upon  34  biscuit,  eating  an  equal  quan- 
tity each  day.     How  many  did  he  eat  each  day? 

16.  What  is  1-tenth  of  34? 

17.  If  a  bar  of  silver,  that  is  worth  37  dollars,  should 
be  cut  into  3  equal  parts,  how  many  dollars  would  1  of 
the  parts  be  worth? 

18.  What  is  1-third  of  37? 

19.  Suppose  a  party  of  9  goldvhunters  find  a  quantity 
of  ore,  which  is  worth  88  dollars;  what  is  the  value  of 
each  man's  share? 

20.  What  is  1-ninth  of  88? 


21.       FRACTIONS  AND  RELATIONS.        69 

21.  If  It  take  a  man  4  months  to  earn  38  dollars,  how 
much  does  he  earn  in  1  month  ? 

22.  What  Is  1 -fourth  of  38? 

23.  If  6  barrels  of  superfine  flour  cost  35  dollars, 
what  is  the  price  of  1  barrel  of  it  ? 

24.  What  is  1-sixth  of  35? 

25.  Suppose  39  bushels  of  corn  to  grow  upon  1 
acre; — how  much  corn  will  1 -fifth  of  an  acre  produce  ? 

2b'.  What  is  1-fifthof  39? 

27.  If  2  dollars  will  pay  for  13  pounds  of  butter,  how 
many  pounds  can  be  bought  for  1  dollar? 

28.  What  is  1-half  of  13? 

29.  If  8  dollars  will  pay  for  78  pounds  of  cheese,  how 
many  pounds  will  1  dollar  pay  for? 

30.  What  is  1 -eighth  of  78? 

31.  Suppose  10  men  drink  55  gallons  of  beer  in  a 
month; — how  much  will  1  man  drink  in  a  month? 

32.  What  is  1-tenthof  55? 

33.  Suppose  7  acres  of  land  to  produce  60  dollars' 
worth  of  hay; — what  is  the  value  of  the  hay  which  1 
acre  of  the  land  produces? 

34. .What  is  1-seventh  of  60? 

35.  If  1  man  can  clear  a  piece  of  wood-land  in  29 
days,  in  what  number  of  days  would  5  men  clear  it? 

Instruction.  Consider  that  5  men  can  do  5  times  as 
much  work  in  a  day,  as  1  man  can  do:  consequently,  it 
will  take  5  men  only  1 -fifth  of  the  time  that  it  will  take 
1  man  to  clear  the  land. 

36.  How  many  days  will  it  take  7  men  to  do  a  piece 
of  work,  that  1  man  can  do  in  46  days? 

37.  If  1  man  will  drink  a  firkin  of  beer  in  50  days, 
how  many  days  will  it  last  6  men? 

38.  Suppose  24  men  can  hoe  a  piece  of  corn  in  1  day; 
what  number  of  men  must  be  employed  to  hoe  it  in  8  days  ? 

Suggestion.  Each  man,  that  shall  be  employed,  can 
do  8  times  as  much  work  in  8  days,  as  he  can  in  1  day. 

39.  If  40  men  can  build  a  wall  in  1  day,  what  number 
of  men  must  be  employed  to  build  it  in  4  days? 

40.  If  a  cistern  can  be  discharged  by  1  lancet  m  19 
hours,  in  what  time  can  it  be  discharged  by  3  faucets? 


70  ORAL    ARITHMETIC.  VL 

Section   22. 

1 .  If  a  smith  can  make  5  cups  from  12  ounces  of  silver, 
how  much  silver  is  required  to  make  3  cups? 

Direction.  First  find  how  much  silver  would  make 
1  cup;  then,  3  times  that  quantity  would  make  3  cups. 

2.  What  is  3  times  1-fifth  of  12? 

Solution.  1 -fifth  of  12  is  2  and  2-fifths.  3  times  2 
is  6;  3  times  2-fifths  is  6-fifths,  or  1  and  1-fifth.  Then 
6  plus  1  and  1-fifth  is  7  and  1-fifth. 

3.  If  22  bushels  of  wheat  will  make  4  barrels  of  flour, 
how  much  wheat  will  make  6  barrels  of  flour? 

4.  What  is  6  times  1-fourth  of  22? 

5 .  Suppose  the  equipments  for  8  soldiers  to  cost  75  dol- 
lars; what  would  be  the  expense  of  equipping  5  soldiers? 

6.  What  is  5  times  1 -eighth  of  75? 

7.  If  29  tons  of  hay  will  keep  9  horses  through  the 
winter,  how  many  tons  would  6  horses  require? 

8.  What  is  6  times  1-ninth  of  29? 

9.  Suppose  7  acres  of  pasturage  to  be  worth  65  dol- 
lars; what  is  3  acres  of  the  same  pasturage  worth? 

10.  What  is  3  times  1-seventh  of  65? 

11.  If  8  acres  of  pasturage  will  keep  35  sheep,  how 
many  sheep  would  be  sufficient  for  6  acres  ? 

12.  What  is  6  times  1-eighth  of  35? 

13.  Suppose  a  man  to  eat  50  pounds  of  beef  in  8 
weeks;  what  number  of  pounds  would  he  eat  in  9  weeks? 

14.  What  is  9  times  1-eighth  of  50? 

15.  If  it  take  36  yards  of  broad-cloth  to  make  10  suits 
of  clothes,  how  many  yards  would  make  4  suits? 

16.  What  is  4  times  1-tenth  of  36? 

17.  A  trader  gave  59  dollars  for  9  barrels  of  flour,  and 
sold  3  barrels  of  it,  at  the  same  price  per  barrel  that  he 
gave.     For  how  much  did  he  sell  the  3  barrels? 

18.  What  is  3  times  1-ninth  of  59? 

19.  If  6  pounds  of  brown  sugar  be  sold  for  52  cents, 
what  would  be  the  price  of  5  pounds  of  it? 

20.  What  is  5  times  1-sixth  of  52? 

21.  If  8  scholars  use  18  quires  of  paper  in  a  month, 
how  many  quires  would  10  scholars  use  in  a  month? 

22.  What  is  10  times  1-eighth  of  18? 


^.   23  FRACTIONS    AND    RELATIONS.  71 

23.  Suppose  a  stage  to  run  58  miles  in  7  hours;  what 
distance  does  it  run  in  6  hours  ? 

24.  What  is  6  times  j  -seventh  of  58  ? 

25.  If  a  mill  grind  17  bushels  of  corn  in  2  hours,  how 
many  bushels  will  it  grind  in  7  hours  ? 

26.  What  is  7  times  1-half  of  17  ? 

27.  Albert  paid  61  cents  for  9  writing-books,  and 
William  bought  7  writing-books,  paying  at  the  same  rate. 
How  much  did  Wilham's  books  cost  him  ? 

28.  What  is  7  times  1 -ninth  of  61  ? 

29.  Suppose  a  hunter  gets  8  pounds  of  gunpowder  ui 
exchange  for  44  pounds  of  venison;  how  many  pounds 
of  venison  must  he  give  for  10  pounds  of  powder  ? 

30.  What  is  10  times  1-eighth  of  44  ? 

Section  23. 

1.  When  writing  paper  is  sold  at  20  cents  a  quire, 
what  is  the  price  of  1 -third  of  a  quire  ? 

2.  If  1 -third  of  a  quire  of  paper  is  worth  6  cents  and 
2-thirds,  what  is  2-thirds  of  a  quire  worth  ? 

3.  What  is  1 -third  of  20  ?     2-thirds  of  20  ? 

4.  Suppose  a  yard  of  ribbon  to  be  worth  23  cents; 
what  is  1 -fourth  of  a  yard  worth  ? 

5.  If  1 -fourth  of  a  yard  of  ribbon  is  worth  5  cents 
and  3-fourths,  what  is  2-fourths  of  a  yard  worth  ? 

Solution.  2-fourths  of  a  yard  is  worth  2  times  5 
cents  and  3-fourths.  2  times  5  cents  are  10  cents;  2 
times  3-fourths  of  a  cent  are  6-fourths  of  a  cent,  or  I 
cent  and  2-fourths.  10  cents  plus  1-cent  and  2-fourths, 
are  11  cents  and  2-fourths. 

6.  What  is  1-fourth  of  23  ?     2-fourths  of  23  ? 

7.  Suppose  a  pound  of  white  sugar  to  be  worth  23 
cents;  what  is  1 -fifth  of  a  pound  worth  ? 

8.  If  1 -fifth  of  a  pound  of  sugar  is  worth  4  cents  and 
3-fifths,  what  is  3-fifths  of  a  pound  worth  ? 

9.  What  is  1 -fifth  of  23  ?     3-fifths  of  23  ? 
Solution.     1-fifth  of  23  is  4  and  3-fifths 3-fifths 

of  23  is  3  times  4  and  3-fifths.  3  times  4  is  12:  3  times 
3-fifths  are  9-fifths,  or  1  and  4-fifths.  12  plus  1  and  4- 
fiftbs  is  13  and  4-fifths. 


72  ORAL    ARITHMETIC.  VL 

10.  Suppose  a  man  can  walk  34  miles  in  a  day,  what 
distance  can  he  walk  in  1 -sixth  of  a  day  ? 

11.  If  a  man  walk  5  miles  and  4-sixths5  in  1-sixth  of 
a  day,  how  far  will  he  walk  in  5-sixths  of  a  day  ? 

12.  What  is  1-sixth  of  34  ?     5-sixths  of  34  ? 

13.  Suppose  a  bushel  of  corn  to  be  worth  65  cents; 
what  is  1 -seventh  of  a  bushel  worth  ? 

14.  If  1-seventh  of  a  bushel  of  corn  cost  9  cents  and 
2-sevenths,  what  will  3-sevenths  of  a  bushel  cost? 

15.  What  is  1-seventh  of '65  ?     3-sevenths  of  65  ? 

16.  Suppose  1  dollar  will  pay  for  35  pounds  of  rice; 
how  much  rice  will  1 -eighth  of  a  dollar  buy  ? 

17.  If  1-eighth  of  a  dollar  will  buy  4  pounds  and  3- 
eighths,  how  much  will  5-eighths  of  a  dollar  buy  ? 

18.  What  is  1-eighth  of  35  ?     5-eighths  of  35  ? 

19.  Suppose  a  man  earns  70  cents  a  day;  how  much 
does  he  earn  in  1 -ninth  of  a  day  t 

20.  If  a  man  can  earn  7  cents  and  7-ninths,  in  1 -ninth 
of  a  day,  how  much  can  he  earn  in  8-ninths  of  a  day  ? 

21 .  What  is  1 -ninth  of  70  ?     8-ninths  of  70  ? 

22.  Suppose  an  acre  of  land  will  produce  43  bushels 
of  oats;  what  will  1-tenth  of  an  acre  produce  ? 

23.  If  1-tenth  of  an  acre  produce  4  bushels  and  3- 
tenths,  what  will  7-tenths  of  an  acre  produce? 

■    24.  What  is  1-tenth  of  43  ?     7-tenths  of  43  ? 

25.  If  a  yard  of  cloth  will  pay  for  30  pounds  of  cheese, 
how  many  pounds  will  3-fourths  of  a  yard  buy? 

Direction,  First  find  hov/  many  pounds  1-fourth  of 
a  yard  will  pay  for. 

26.  A  farmer  sold  4-fifths  of  a  ton  of  hay,  for  oats, 
allowing  32  bushels  of  oats  to  be  worth  the  same  as  a 
ton  of  hay.     How  many  bushels  of  oats  did  he  receive  ? 

27.  What  is  4-fifths  of  32  ? 

28.  Suppose  1  dollar  will  pay  for  38  pounds  of  rice; 
for  how  manv  pounds  will  8-tenths  of  a  dollar  pay  ? 

29.  What'is  8-tenths  of  38  ? 

30.  A  man  bought  a  piece  of  land  containing  1  acre 
and  4-sixths,  and  paid  at  the  rate  of  40  dollars  per  acre. 
How  much  did  he  pay  for  the  land  ? 

31.  What  is  40  plus  4-sixths  of  40  ? 


34  FRACTIONS   AND  RELATIONS.  78 

Section  24. 

1.  Suppose  3-fourths  of  a  yard  of  flannel  to  cost  S2 
c«nts;  what  does  1 -fourth  of  a  yard  cost  ?  What  would 
a  yard  cost  ? 

Solution,  If  3-fourths  of  a  yard  cost  32  cents,  1- 
fourth  of  a  yard  costs  1 -third  of  32  cents.     1 -third  of 

32  cents  is  10  cents  and  2-thirds  of  a  cent 4-fourths, 

or  a  whole  yard  would  cost  4  times  10  cents  and  2-thirds. 
4  times  10  cents  are  40  cents;  4  times  2-thirds  are  8- 
thirds,  equal  to  2  and  2-thirds.  Then,  40  cents  plus 
2  cents  and  2-thirds  are  42  cents  and  2-thirds. 

2.  32  is  3-fourths  of  what  number  ? 

Solution.  Since  32  is  3-fourths  of  the  number,  1- 
third  of  32  is  1-fourth  of  it.  1 -third  of  32  is  10  and 
2-thirds.     4  times  10  and  2-thirds  are  42  and  2-thirds. 

3.  If  2-fifths  of  an  acre  of  land  will  produce  9  bushels 
of  rye;  how  many  bushels  will  1-fifth  of  an  acre  produce  ^ 
How  many  bushels  will  an  acre  produce  ? 

4.  9  is  2-fifths  of  what  number  ? 

Instruction.  Observe^  that  1-half  of  9  must  be  1-fifth 
of  the  required  number. 

5.  If  a  man  drink  6  gallons  of  beer  in  5-sixths  of  a 
month,  how  many  gallons  does  he  drink  in  1-sixth  of  a 
month  ?     How  many  galbns  will  he  drink  in  a  month  ^ 

6.  6  is  5-sixths  of  what  number  ? 

7.  A  man,  who  spends  43  cents  a  day,  finds  his  ex- 
penses to  be  5-sevenths  of  his  wages.  What  is  1-sev- 
enth  of  his  wages  ?     What  is  the  whole  of  his  wages  ? 

8.  42  is  5-sevenths  of  what  number  ? 

9.  If  5-eighths  of  a  dollar  will  pay  for  24  pounds  of 
flour,  how  many  pounds  will  1 -eighth  of  a  dollar  pay 
for  ?     How  many  pounds  will  a  dollar  pay  for  ? 

10.  24  is  5-eighths  of  what  number? 

11.  Suppose  6  gallons  of  wine  to  leak  from  a  cask  in 
8-ninths  of  an  hour;  how  much  will  leak  out  in  1-ninlh 
of  an  hour  ?     How  many  gallons  in  1  hour  ? 

12.  6  is  8-ninths  of  what  number  ? 

13.  If  4-tenths  of  a  yard  of  cloth  be  worth  33  cents, 
how  much  is  1  yard  worth  ? 

Direction,     First  find  what  I  •tenth  of  a  yard  is  worth- 


74  ORAL   ARITHMETIC.  VL 

14  If  3-eighths  of  a  bale  of  cotton  be  worth  17  dol- 
lars, what  is  the  whole  bale  worth  ? 

15.  A  laborer  spent  30  cents  a  day,  and  still  saved 
3-sevenths  of  his  wages.     How  much  was  his  wages  ? 

16.  Suppose  that  I  have  read  5-ninths  of  the  pages  in 
a  certain  book,  and  there  are  35  pages  more  to  be  read; 
— how  many  pages  are  there  in  the  book  ? 

NOTATION    OF    FRACTIONS. 

Learners  will  now  attend  to  the  meaning  of  the  words, 

FRACTION,  DENOMINATOR,  and  NUMERATOR. 

A  Fraction  is  any  part  of  one.  For  example,  one-half 
of  an  orange  is  r  fraction  of  1  orange;  three-fourths  of 
an  orange  is  Ruoiher  fraction  of  1  orange. 

In  this  book,  fractions  have  been  expressed  by  a 
number  joined  with  a  word;  thus,  4-ninths.  Fractions 
are  commonly  expressed  by  two  numbers,  standing  one 
above  the  other,  with  a  line  between  them;  thus,  ^  h^f\ 

1    one-  2    t^wo-  I    one-  4    four-  3    three- 

"3"  third,        "3    tliirds,         "6    sixth,  5"   fifths,         9    ninths. 

17.  What  fraction  is  expressed,  when  there  is  a  4  with 
a  1  over  it. '^  7  with  2  over  it?  8  with  5  over  it.-^  10 
with  6  over  it  ? 

18.  Which  is  the  greater  fraction;  ^  or  ^?  ^  or  ^.^ 
Y  01  -^  .''     g^  01  10  r     4  or  Q  r 

19.  Which  is  the  greater  fraction;  ^  or  f  ?     f  or  f  ? 

3   or    1   ?        4   ^«   7  ?         8     ^^  _3_  ? 

y  or  5  r     -Q  or  ^  r     -^q  or  ^q  . 

The  Denominator  of  a  fraction,  is  the  number  of  equal 
parts  into  which  a  whole  one  is  divided.  For  example, 
if  a  whole  orange  be  divided  into  4  equal  parts,  the  de- 
nominator is  4;  the  parts  being  denominated /okW/i.9. 

The  JsTximerator  of  a  fraction  is  the  number  which 
shows  how  many  of  the  equal  parts  the  fi'action  ex- 
presses. For  example,  the  fraction  f  expresses  3  of 
the  four  equal  parts;  therefore  3  is  the  numerator. 

20.  What  numerator,  and  what  denominator,  would 
express  the  fraction,  four-fifths  ?  two-eighths  }  six- 
ninths  .^     one-fifteenth.^     five-eighteenths.^ 


•35 


FRACTIONS. 


75 


When  the  numerator  is  equal  to  the  denominator, 
thus,  I,  then  the  fraction  is  equal  to  1;  as  4 -fourths  of 
an  orange,  when  joined  together  make  1  orange. 

When  the  numerator  is  greater  than  the  denominator, 
thus,  I,  then  the  fraction  is  equal  to  as  many  times  1  as 
the  denominator  is  contained  times  in  the  numerator. 

21.  How  many  times  1,  [how  many  whole  ones], in |  ? 


m  V'  ? 


20   ? 


30  ? 

To  ' 


24  ? 
12  • 


22.  How  many  times  1,  and  what  fraction  over,  in  |  ^ 
I?     inV?     in¥?     in  II? 

23.  Where  have  you  observed  the  numerator  of  a  frac- 
tion to  stand; — above^  or  below  the  denominator  ? 


mf? 


Section  25, 

1.  James  has  ^  of  a  dollar,  and  Henry  has  f  of  a 
dollar: — which  of  them  has  the  most  money  ? 

Compare   the   fraction    ^ 
with  other  fractions. 

2.  ^  is  equal  to  how  ma- 
ny fourths  } 


how  many  six 


3.  ^  is  equal  to  how  ma- 
ny sixths  } 

4.  i  is  equal  to  how  ma- 
ny eighths  ? 

5.  ^  is  equal  to  how  ma- 
ny tenths  ? 

6.  ^  is  equal  to  how  many  twelfths  ? 
teenths  ?     how  rnany  twentieths  ? 

7.  Edward  broke  a  slate  pencil  into  3  equal  pieces, 
and  Albert  broke  one  into  6  equal  pieces.  How  many 
of  Albert's  pieces  were  equal  to  1  of  Edward's  pieces  ? 

Compare   the   fraction  ^ 
with  other  fractions. 

8.  ^  is  equal  to  how  ma- 
ny sixths  ? 

9.  ^  is  equal  to  how  ma- 
ny ninths  ? 


1       ^m 

1 , 1 

• 

i     HH 

:    ^^m 

■■  i   m^m 

76  ORAL    ARITHMETIC.  Vl. 

1§.  -3-  is  equal  to  how  many  twelfths  ?  how  many 
eighteenths  ?     how  many  thirtieths  ? 

Suggestion,     y  of  1  is  equal  to  ^  of  12-twelfths. 

11.  ^  is  equal  to  how  many  eighths  ?  how  many 
twelfths  ?     how  many  sixteenths  ? 

12.  I  is  equal  to  how  many  tenths  ?  how  many  twen- 
tieths ?     how  many  twenty -fifths  ? 

13.  i  is  equal  to  how  many  twelfths  ?  how  many 
eighteenths  ?     how  many  thirtieths  ? 

14.  ^  is  equal  to  how  many  fourteenths.^  how  many 
twenty-eighths  ?     how  many  thirty-fifths  ? 

15.  I  is  equal  to  how  many  twentieths  ? 

Solution,  ^  is  equal  to  ^05  I  ^^  equal  to  3  times  5- 
twentieths,  which  is  ^f . 

16.  f  is  equal  to  how  many  twelfths  ? 

17.  A  boy,  who  had  f  of  an  orange,  cut  each  fifth 
mto  2  parts,  (making  tenths);  his  brother  gave  him 
^  more.     What  fraction  of  an  orange  had  he  at  last  ? 

18..  Into  how  many  parts  must  you  cut  a  sixth  of  an 
orange,  to  make  the  parts  eighteenths? Why  .'^ 

19.  f  is  equal  to  how  many  eighteenths  ? 

20.  Change  f  to  fourteenths,  and  then  add  ^  to  it. 

21.  1^  is  equal  to  how  many  twenty-fourths  ? 

22.  Change  f  to  eighteenths,  and  then  take  -^  from  it. 

Section   26. 

1.  What  is  meant  by  a  Fraction  ?* How  is  a  frac- 
tion commonly  expressed  ?. What  is  the  Denomina" 

tor  of  a  fraction  ?. What  is  the  JSTumerator  ? 

2.  If  the  denominator  of  a  fraction  be  9,  and  the 
numerator  7,  how  should  these  numbers  be  written  ? — • 
and  what  would  the  fraction  be  called  in  reading  it  ? 

3.  Suppose  two  fractions  have  numerators  alike,  and 
denominators  different — which  is  the  greater  fraction — 
that  with  the  greater,  or  the  smaller  denominator  ? 

4.  Suppose  two  fractions  have  denominators  alike, 
and  numerators  different — which  is  the  greater  fraction 
— that  with  the  greater,  or  the  smaller  numerator  ? 


26.  27  FRACTIONS.  77 

Observation.  If  an  orange  be  cut  into  eighths,  and 
then  4  of  the  eighths  be  joined  together,  these  A-eighths 
become  l-Zia/fof  an  orange.  And  thus  the  fraction,  |, 
when  reduced  to  its  lowest  terms,  is  ^. 

5.  Reduce  f  to  its  lowest  terms — that  is,  find  the 
lowest  numerator  and  denominator,  that  will  express  a 
quantity  equal  to  f . 

6.  Reduce  |  to  its  lowest  terms. 

7.  Reduce  f  t«  its  lowest  terms.     Reduce  f. 

8.  Reduce  f  to  its  lowest  terms.     Reduce  |-. 

9.  Reduce  |  to  its  lowest  terms.     Reduce  f. 
Observation,     A  fraction  is  reduced,  by  dividing  the 

numerator  and  denominator  by  any  number,  which  will 
divide  them  both  without  a  remainder  For  example, 
we  reduce  -^q  thus;  2  in  6,  3  times,  3  is  a  new  numera- 
tor; 2  in  10,  5  times,  5  is  a  new  denominator. 

10.  Reduce  each  of  the  following  fractions  to  its  low- 

^cf  forme  5  4  8  6  „4_  8         _2_         1_0 

est  terms,    -jq.    -jq.    -yq-    12\    12'    T5^-    12-    is- 

11.  Stephen's  knife  cost  -^^  ^^  ^  dollar,  and  John's 
cost  -^Q  of  a  dollar.     Whose  knife  cost  the  most } 

12.  Reduce  the  fractions,  /^  ^"^  io  ^^  their  lowest 
terms,  and  then  add  them  together. 

13.  Reduce,  and  then  add  together,  -^§  and  |f . 

14.  Reduce,  and  then  add  together,  If  and  2%- 

Section  27. 

1.  -4-  of  a  water  melon  was  divided  equally  among-  3 
boys.     What  part  of  the  whole  melon  did  1  boy  receive  } 

2.  -J    of  :^  is  equal  to 


I 


what  part  of  1 } 

Solution.  If  \  be  divided  into  3  equal  parts,  it  will 
take  12  such  parts  to  make  a  whole  one.     Therefore, 

s- of?  is  ^^2-     [^times  4  isy^-l 

3.  What  part  of  a  whole  one  is  ^  of  |  ? 

Illustration.  If -J-  of  an  orange  were  cut  into  6  equa. 
parts,  it  would  take  5  times  6  such  parts  to  make  a 
whole  orange.     Operation,     -g^  times  ^  is  y^ . 


78  ORAL    ARITHMETIC.  VI. 

4.  What  part  of  1  is  -^  of  ^  ?     i  of  -»-  ?     \  of  -J  ?    i  of 
i?    iofi?     iof^? 

5.  If  18^  dollars  [18  and  |  dollars]  be  divided  equal 
ly  among  3  men,  what  will  each  man  receive  ? 

6.  What  is  I  of  18i? 

Sol,  ^  of  18  is  6;  -j  of  ^  is  y^^;  6  plus  -^^  is  6-^^^. 

7.  Whatis  jof  30-^?     iof24i?     jofl8-}? 

8.  A  boy,  having  }j  of  a  dollar,  paid  -|  of  his  money 
for  a  knife.     What  part  of  a  dollar  did  he  pay  ? 

9.  I   of  'I   is    equal  to 


what  part  of  1  ? 

Solution,      One-fourth  of  ^  is  equal  to  ^;  ^/irce-fourths 
of  ^  is  equal  3  times  I,  which  is  f . 

10.  Whatpartof  lisfofi.?     |of|.?     |ofl.?     fof 

1  ?      5  p.r  1  p      6  ^f   1    ? 

^  r     y  01  -g-  r     g  01  1  (^  r 

11.  A  girl  having  |-  of  a  dollar,  paid  -^  of  her  money 
for  a  book.     What  part  of  a  dollar  did  she  pay  ? 

12.  I  of  I  is   equal  to 
what  part  of  1  ? 

Solution,     l  of  one-fourth  [^  tin^^s  -^  is  -q]  is  equal  to 
^;  ^  of  I  is  3  times  i,  which  is  |- . 

1 3.  Which  is  the  greater  fraction  of  a  dollar, — -|  of  I 
of  a  dollar, —  or,  -^  of  |  of  a  dollar  ? 

14.  What  part  of  1  is  i  of  f  ?     i  of  |  ?     -^of|?     -\ 
of  3?     iof|?     lof-/^? 

15.  If  4  cloaks  are  to  be  made  from  12 f  yards  of 
cloth,  how  many  yards  must  be  put  into  each  cloak  ? 

16.  Whatis  iof  12|? 

Solution,     ^  of  12  is  3:    ^l^  of  ^  is  -^^ ,  -}  of  f  is  ^2  • 
llien  3  plus  ^2  ^^  ^A* 

17 .  What  is  i  of  20f  ?     -}  of  28 1  ?     ^  of  45 1  c 

p     18.  A  boy  having  f  of  a  dollar,  paid  -|  of  his  money 
for  a  book.     What  part  of  a  dollar  did  he  pay  ? 

19.  I  of  f  is  equal  to 
what  part  of  1  ? 

Solution,     'I  of  I-  is  equal  to  -^^  5  4  ^^  i  ^s  -^2  5  I  ^f  f 
is  2  times  ^^ ,  or  ^^ .     -f^  is  equal  to  ^ . 


28.  FRACTIONS  79 

20.  f  of  an  acre  of  land  was  divided  into  5  equal  lots, 
and  a  gardener  bought  3  of  the  lots.  What  part  of  an 
acre  did  he  buy  ? 

Direction.  First  find  what  part  of  an  acre  there  was 
in  one  lot; — then,  what  part  in  3  lots. 

21.  What  part  of  1  is  |off  ? 

Direction.  First  find  what  part  of  a  w^hole  one  \  of 
I  is; —  then  find  j  of  f ,  —  and  then  f  of  f  . 

22.  What  part  of  1  is  f  off  ?  |of-|.?  |of|.?  f  of 
f?     foff.^     fof|.? 

23.  A  merchant,  who  owned  |  of  a  ship,  sold  f  of 
his  share.     What  part  of  the  ship  did  he  sell  ? 

24.  Whatpartof  lis|of|.?     fof|.J^     F^ff?     |  of 

T  •         8    "*    8  •         5   ^^  To  • 

25.  Suppose  a  piece  of  broad-cloth  to  contain  32| 
yards; — how  many  yards  are  there  in  |  of  the  piece  ^ 

Direction.     First  find  |  of  32;  then  find  |  of  |. 

26.  Whatisf  of20f .?     fof36|.?     i%of40f .?»     fof 

Sof?     fof54|-.?     -/oOfSOf.? 

Section   28. 

1.  Suppose  you  have  -3-  of  an  orange  and  \  of  an  or- 
ange,—  into  how  many  parts  must  you  cut  the  third ^ 
and  into  how  many  parts  the  fourth^  so  that  the  parts  of 
the  third,  and  of  the  fourth  shall  be  of  equal  size  ? 

We  here  see,  that  when 
l  is  divided  into  4  parts, 
and    \-   into    3   parts,    the 


parts  are  all  twelfth 

In  this  example  12  is  found  to  be  a  Common  Denomi- 
nator; and  the  two  fractions  -5-  and  \ ,  become  -^^  and  -^^ . 

2.  Change  \  and  -^  to  a  common  denominator:  that  is, 
find  how  many  parts  a  half^  and  how  many  a  third  must 
be  divided  into,  so  that  the  parts  shall  be  equal:  also 
find  how  many  of  these  parts  would  make  a  whole  one. 

Observation.  If  two  denominators  be  multiplied  to- 
gether they  will  produce  a  common  denominator. 


80  ORAL    ARITHMETIC.  VI 

3.  Change  \  and  |  to  a  common  denominator. 
Solution.     3  times  5  is  15,  the  common  denominator 

^  of  -i|  is  iV  ;    i  of  -^1  is  -^j .     Answer,  -^j  and  i\ . 

4.  Change  \  and  ^  to  a  common  denominator, 

5.  Change  ^  and  ^  to  a  common  denominator. 

6.  Change  ^  and  3^  to  a  common  denominator. 

7.  Change  ^  and  :5^  to  a  common  denominator. 

8.  Change  y  and  |^  to  a  common  denominator.    ^ 

9.  Change  |^  and  \  and  |  to  twelfths. 

10.  Change  -I  and  ^  and  |-  to  twenty-fourths. 

11.  Change  \  and  -5^0  and  xs  to  thirtieths. 

12.  Change  ^^  and  f  to  a  common  denominator. 
Solution,     4  times  5  is  20,  the  common  denominator 

4  "^   20  ^^  20  '    y  ^^  20  ^^  20^  5"  ^^  '^  umes  20?  so- 
ls.  Change,^  and  |  to  a  common  denominator. 

14.  Change  f  and  ^  to  a  common  denominator.  .•  ' 

15.  Change  |  and  -5-  to  a  common  denominator. 

16.  Change  ^  and  y  to  a  common  denominator. 

17.  Change  f  and  f  to  a  common  denominator. 

18.  Change  f  and  y  to  a  common  denominator. 

19.  Change  -^  and  -f  to  a  common  denominator. 

20.  Change  |-  and  |-  to  a  common  denominator. 

21.  How  much  is  f  and  f  added  together. 
Solution,     ^  is  equal  to  ^V?  ^"^  f  is  ^f  :    y  is  equal  to 

jV  and  f  is  if.     ^1  plus  ^f  is  ff ,  equal  to  ^|. 

22.  How  much  is  \  and  4  added  too;ether  ? 


23.  How  much  is  f  and  f  added  together 


24.  How  much  is  f  and  f  added  together  ? 

25.  How  much  is  f  and  |  added  together  ? 
26    How  much  is  f  and  |  added  together  ? 

27.  ^  and  \  and  -j^^  are  how  many  twelfths  ? 

28.  f  and  ^  and  -j^  are  how  many  twelfths  ? 

'  29.  :^  and  j  and  -^j  are  how  many  sixteenths  ?  * 

30.  If  4  be  taken  from  f ,  how  mudi  will  remain  .•* 
Solution,      [y  times  y  is  ^5]  .  y  is  equal  to  ^ ,  |  is  |^  . 

^  is  equal  to  ^,  |  is  f}.     Then  H  minus  ff  is  ^. 


29  FRACTIONS.  M 

53.  Take  f  from  |,  —  how  much  remains  ? 

34.  Take  f  from  1^, — how  much  remains  ? 

35.  Take  f  from  |, — how  much  remains  ? 

36.  Take  J  from  f,  —  how  much  remains  ? 

37.  Take  f  from  ^, — how  much  remains  ? 

SECTION    29. 

1.  A  farmer  gathered  21  f  bushels  of  apples  from  one 
tree,  and  lOf  from  another.  How  many  bushels  did  he 
gather  from  both  trees  ?  r 

Direction.  First  add  together  the  whole  bushels, 
then  change  the  fractions  of  a  bushel  to  a  common  de- 
nominator and  add  the  new  numerators. 

2.  If  a  bonnet  cost  5|  dollars  and  a  shawl  5-{q  dollars, 
how  much  do  they  both  cost  ? 

3.  On  a  certain  day,  I  travelled  30|  miles  in  a  stage, 
154  miles  in  a  gig,  and  10  miles  on  horseback.  How 
many  miles  did  I  travel  that  day  ? 

4.  A  farmer  sold  a  cow  for  23|-  dollars,  and  a  calf  for 
4 1  dollars.     How  much  did  he  get  for  both  ? 

5.  Three  soldiers  shared  a  loaf  of  bread  as  follows  :-^- 
the  first  man  took  f  of  it,  the  second  took  ^  of  it,  and 
the  third  took  the  remainder.  What  part  of  the  loaf  did 
the  third  soldier  get  ? 

6.  Three  men.  A,  B,  and  C,  are  to  reap  a  field  of 
wheat — A  is  to  reap  |  of  it,  B  ^%  of  it,  and  C  the  re- 
mainder.    What  part  of  the  field  is  C  to  reap  ^ 

7.  A  trader,  having  2  barrels  of  flour,  sold  |  of  a 
barrel  to  one  man,  and  f  of  a  barrel  to  another  man. 
What  part  of  a  barrel  had  he  remaining  ? 

8.  A  man,  having  10  dollars,  paid  away  4\  dollars 
for  a  hat,  and  3^  dollars  for  a  pair  of  boots.  How 
many  dollars  had  he  left  ? 

9.  A  miller,  having  20  bushels  of  corn,  sold  6f  bush- 
els to  one  man,  and  9|-  to  another.  How  many  bushels 
had  he  remaining  ? 

10.  A  man  paid  25  f  dollars  for  a  watch,  and  2^^^  dol- 
lars for  having  it  repaired,  and  then  sold  it  so  as  to  gain 
8  dollars.     For  how  much  did  he  sell  it.^ 


est  ORAL    ARITHMETIC.  VI. 

Section  30. 

1.  Suppose  I  had  4  oranges, — to  how  many  boys 
could  I  give  f  of  an  orange  apiece  ? 

Direction,  Find  how  many  thirds  of  1  orange  in  4 
oranges,  then  find  how  many  times  2-thirds  there  are. 

2.  How  many  times  is  f  contained  in  4  ? 
Solution.     1  is  equal  to  |,  and  4  is  equal  to  4  times 

I  or  ^y  :  then  f  is  contained  in  ^-^ ,  6  times. 

3.  How  many  pairs  of  gloves  can  I  buy  for  6  dollars, 
the  price  being  f  of  a  dollar  a  pair  ? 

4.  How  many  times  is  |  contained  in  6  ? 

5.  Suppose  a  man  to  walk  1  mile  in  |  of  an  hour, — 
what  distance  will  he  walk  in  1  hour  ? 

6.  How  many  times  is  f  contained  in  1  ? 
Solution.     1  is  equal  to  |-.     f  in  -I,  4^  times. 

7.  How  many  yards  of  cloth,  that  is  sold  for  f  of  a 
dollar  a  yard,  can  be  bought  for  4  dollars  ? 

8.  How  many  times  is  f  contained  in  4  ? 

9.  How  many  pounds  of  tea,  that  is  sold  for  f  of  a 
dollar  a  pound,  can  be  bought  for  4-^  dollars  ? 

10.  How  many  times  is  f  contained  in  4^? 

11.  If  I  of  a  barrel  of  biscuit  will  last  a  ship's  crew  1 
week,  how  many  weeks  will  3 1  barrels  last  them } 

12.  How  many  times  is  y  contained  in  3f  ? 

13.  How  many  yards  of  cloth,  at  ^  of  a  dollar  per 
yard,  can  be  bought  for  |  of  a  dollar  ? 

Solution,  -J  of  a  dollar  is  equal  to  -^j  of  a  dollar;  | 
of  a  dollar  is  equal  to  -jf  of  a  dollar.  As  many  yards 
can  be  bought  as  -^^  is  contained  times  in  ^. 

14.  How  many  times  is  ^  contained  in  |  ? 

15.  If  a  man  can  hoe  -j  of  a  field  of  corn  in  1  day,  in 
how  many  days  can  he  hoe  -J  of  the  field  ? 

16.  How  many  times  is  ^  contained  in  |  ? 

17.  How  many  times  is  f  contained  in  f  ? 
Direction.     Change  both  fractions  to  a  common  de« 

nominator;  then  divide  one  numerator  by  the  other 

18.  How  many  times  is  f  contained  in  ^} 

19.  How  many  times  is  f  contained  in  ^  ^ 

20.  How  many  times  is  |  contained  in  -^^  ? 


30.  31.  •  FRACTIONS.  83 

21 V  Suppose  that  6  cloaks  are  to  be  made  from  22| 
.yards  of  broad-cloth; — what  number  of  yards  must  be 
put  into  each  cloak  ? 

Solution,  Each  cloak  must  contain  -g^  of  22 1  yards. 
\  of  22|  is  3,  there  being  4f  over.  4|  is  equal  to  y  . 
^of  \  is  2V  9  ^^^  i  ^^  V  is  1^  times  as  much,  or  ^|. 
Then  3  yards  plus  H  of  a  yard,  are  3J|  yards. 

22.  If  30f  pounds  of  bread  will  supply  a  family  for  1 
week,  how  many  pounds  will  supply  the  family  for  1  day  : 

23.  What  is  ^of  30f  ? 

24.  If  8  yards  of  cloth  cost  51  f  dollars,  what  will  1 
yard  cost }     What  will  3  yards  cost } 

25.  What  is  I  of  51 1  ?     What  is  |  of  51  f .? 

Section  31* 

Note  to  Teachers,  Thissectionfurnishesatestof  the  learner's  knowledge 
of  the  several  operations  taught  since  the  Review  in  Section  12.  Should  tlie 
learner  fail  in  any  of  these  examples,  he  must  be  put  back  to  the  section,  whose 
number  is  prefixed  to  the  example  in  whicli  tlie  failure  appears. 

REV  IE  W. 

1.  (§  13.)  How  many  hom's  will  it  take  you  to  read 
a  book  of  75  pages,  if  you  read  9  pages  an  horn*  }    •  , 

2.  ( §  14.)  If  a  bushel  of  oats  be  given  in  exchange  for 
I  of  a  bushel  of  grass  seed,  how  many  bushels  of  oats 
must  be  given  for  (>|  bushels  of  grass  seed  ? 

3.  (§  15.)  If  a  man  drink  \  of  a  gallon  of  beer  in  a 
day,  how  many  gallons  will  he  drink  in  33  days } 

4.  (§  16.)  Suppose  a  watch  to  cost  17 1-  dollars,  and 
a  chain  1 1-  dollars, —  what  is  the  cost  of  both  ^ 

5.  ( §  17.)  If  1  quire  of  letter  paper  cost  y^  of  a  dol- 
lar, what  will  7  quires  cost,  at  the  same  rate  ? 

6.  (§  IS.)  Suppose  a  fire  engine  to  throw  from  its 
pipe,  4 1  barrels  of  water  in  1  minute, —  what  number  of 
barrels  will  it  throw  in  10  minutes  ? 

7.  (§19.)  A  farmer  sold  -}  of  a  ton  of  hay  for  3|- 
dollars.     What  is  the  price  ol  a  ton  at  the  same  rate  ? 

8.  ( §  20.)  There  were  9  men,  who  performed  a  piece 
of  work,  for  which  they  received  6  bushels  of  wheat. 
What  part  of  a  bushel  was  the  share  of  each  man  } 


§4  ORAL   ARITHMETIC.  VI 

9.  (§21.)  A  ship's  crew'used  14  casks  of  water, 
during  a  passage  of  5  months,  from  Calcutta  to  New  York 
How  much  did  that  quantity  allow  them  per  month  ? 

10.  ( §  22.)  If  7  barrels  of  flour  cost  30  dollars,  what 
will  9  barrels  cost  at  the  same  rate  ? 

11.  (§23.)  A  man  purchased  a  farm,  containing  95 
acres;  but  not  being  able  to  pay  for  the  whole,  he  sold 
ofli  Yo  of  the  land.     How  many  acres  did  he  sell  .'^ 

12.  (§  24.)  If  a  mill  grind  9  bushels  of  corn  in  f  of 
an  hour,  how  many  bushels  will  it  grind  in  1  hour  ? 

13.  (§25.)  If  pen-knives  are  worth  -}  of  a  dollar 
apiece,  and  pencils  gV  ^^  ^  dollar  apiece,  how  many 
pencils  must  be  given  in  exchange  for  3  knives  ? 

14.  (§26.)  Reduce  -Jf  to  its  lowest  terms.  How 
do  you  reduce  a  fraction  to  its  lowest  terms  ? 

15.  ( §  27.)  A  man,  owning  -|  of  an  acre  of  land,  sold 
f  of  what  he  owned.     What  part  of  an  acre  did  he  sell  ? 

16.  (§  2S»)  Change  |  and  f  to  a  common  denomina* 
tor.     How  do  you  change  fractions  to  a  com.  denom.  ^ 

'17.  (§  2S.)  John  gave  y  of  a  dollar  for  a  book,  and 
\  of  a  dollar  for  a  slate,  and  then  sold  them  both  for  | 
of  a  dollar.     Did  he  gain  or  lose  ? —  How  much  ^ 

IS.  (§29.)  A  farmer  cut  18|  tons  of  hay,  and  sold 
2f  tons  of  it.     How  many  tons  had  he  left  ? 

19.  (§  30  )  When  coffee  is  f  of  a  dollar  per  pound, 
how  many  pounds  can  be  bought  for  f  of  a  dollar  ? 

20.  (§  30.)  A  tenant  raised  58  |  bushels  of  corn,  and 
gave  his  landlord  -|-  of  it  for  the  use  of  the  land.  How 
many  bushels  had  the  tenant  for  himself.'! 

Section  32. 

MISCELLANEOUS    EXAMPLES. 

1.  The  Gulf  Stream  is  a  current  in  the  ocean,  running 
3  miles  an  hour.  If  a  steam  boat,  whose  engine  propels 
her  12^  miles  an  hour,  should  run  in  the  stream,  with 
the  current,  what  distance  would  it  move  in  8  hours  ? 

2.  If  the  above  steam  boat  were  running  against  the 
current,  what  distancie  would  it  move  in  8  hours  ? 


82.  MISCELLANEOUS  EXAMPLES.  85 

3.  A  trader  bought  25  barrels  of  flour,  paying  7  dol* 
lars  a  barrel  for  1 1  barrels  of  it,  and  9  dollars  a  barrel  for 
the  remainder.     What  did  the  whole  cost  ? 

4.  What  sum  of  money  must  be  divided  among  10 
men,  so  that  each  man  shall  receive  19}  dollars  ? 

5.  Suppose  a  man  can  perform  a  journey  iji  8  days, 
travelling  10  hours  a  day, —  in  how  many  days  can  he 
perform  it,  travelling  12  hours  a  day.'* 

6.  Henry  reads  12  pages  in  the  same  time  that  Wil- 
liam is  reading  7  pages; — how  many  pages  will  Henry 
read  while  William  is  reading  60  ? 

7.  If  72  dollars  be  divided  equally  among  9  sailors, 
how  many  weeks'  board,  at  3  dollars  a  week,  will  each 
sailor's  share  pay  for  } 

S.  A  man  failed  in  trade,  and  could  pay  only  4  dollars 
on  every  9  dollars  that  he  owed.  How  much  did  he 
pay  on  a  debt  of  100  dollars  ? 

9.  There  is  a  pole  standing  in  a  pond,  so  that  |  of  it 
is  under  the  water,  and  3|  feet  of  it  is  above  the  water. 
How  long  is  the  pole  ? 

10.  A  pole  is  standing  so  that  I-  of  it  is  in  the  mud, 
f  of  it  is  in  the  water,  and  2  ^  feet  of  it  is  above  the  wa- 
ter.    What  is  the  length  of  this  pole  ? 

11.  If  A  borrow  of  B,  8  bushels  of  wheat,  when  the 
price  is  9  shillings  a  bushel;  how  much  wheat  must  A 
return,  when  the  price  is  7  shillings  a  bushel  ? 

12.  A  trader,  having  100  dollars,  laid  out  -/q  of  his 
money  for  narrow-cloth  at  5  dollars  a  yard,  and  the 
remainder  for  broad-cloth  at  7  dollars  a  yard.  How 
many  yards  did  he  buy  of  each  sort  ? 

13.  If  2  2"  bushels  of  apples  will  fill  a  barrel,  how  many 
bushels  will  it  take  to  fill  8  barrels  ? 

14.  John  can  pick  a  quart  of  berries  in  an  hour;  Ann 
can  pick  twice  as  fast; — how  many  can  both  pick  in  an 
hour  ?     In  what  time  would  they  pick  10  quarts  ? 

15.  How  many  bushels  of  corn  must  a  miller  grind, 
to  get  1  bushel  for  himself — allowing,  that  he  takes  2 
quarts  from  every  bushel  before  grinding  it.  and,  that  32 
quarts  make  a  bushel  ? 

Suggestion^    He  gets2(|ts.  for  grinding  less  than  a  bush. 


86  ORAL    ARITHMETIC.  VI 

16.  If  1  monitor  can  mend  2  pens  in  a  minute,  how 
long  will  it  take  3  monitors  to  mend  28  pens  ? 

17.  It  is  worth  as  much  to  pasture  1  cow,  as  5  sheep. 
If  I  pay  1  dollar  a  month  for  pasturing  a  cow,  what 
must  I  pay  for  pasturing  35  sheep,  7  months  ? 

18.  If  3  horses  eat  1  ton  of  hay  in  1  month,  how 
long  will  5  tons  last  4  horses  ? 

19.  A  drover  sold  a  cow  for  20  dollars,  and,  in  so 
doing,  he  gained  a  sum  equal  to  -3-  of  what  he  had  paid 
for  the  cow.     How  much  had  he  paid  for  her  ? 

20.  Suppose  a  man  can  dig  a  trench  in  4  days,  and  a 
boy  in  6  days; — what  part  of  it  can  each  dig  in  1  day  .'* 
What  part  of  it  can  both  together  dig  in  1  day }     In 
what  time  can  they  both  finish  it  ? 

21.  Suppose  a  cistern  has  one  tap  that  will  discharge 
it  in  5  hours,  and  another  in  7  hours, —  in  what  time 
will  they  both  discharge  it  ? 

22.  If  1  man  can  perform  a  piece  of  work  in  35  days, 
m  what  time  can  6  men  perform  it  ? 

23.  If  4  men  drink  a  barrel  of  cider  in  20  days,  in 
what  time  will  9  men  drink  the  same  quantity  ? 

24.  If  9  men  can  do  a  piece  of  work  in  5  days,  in 
how  many  days  will  7  men  do  the  same  w^ork  ? 

'  25.  A  farmer  kept  his  sheep  in  four  pastures —  In  the 
first  pasture  he  had  -^^  of  his  flock;  in  the  second,  1%; 
in  the  third,  ^q;  and  in  the  fourth  he  had  32  sheep* 
How  many  sheep  had  he  ? 

26.  There  is  a  school,  in  which  ^  of  the  scholars 
read  in  the  Classical  Reader^  i  read  in  the  JSTational 
Reader^  \  read  in  Plerponfs  Introduction^  and  36  httle 
boys  read  in  the  Young  Reader.  How  many  scholars 
are  there  in  the  school  ? 

27.  If  a  post  4  feet  high  cast  a  shadov/  3  feet,  ai 
noonday,  what  is  the  height  of  a  steeple,  that  casts  a 
shadow  90  feet,  at  the  same  time.   ; 

28.  A  and  B  are  laborers — A  earns  10  dollars  a 
month,  and  B  9;  but  A  gives  \  of  his  earnings  to  B. 
What  will  each  lay  up  in  3  months  .'' 

29.  If  12  men  can  perform  a  piece  of  work  in  6  days, 
m  w*hat  time  would  10  men  perform  the  work  ? 


82.  MISCELLANEOUS    EXAMPLES.  87 

30.  How  many  men  must  be  employed,  to  dig  a 
trench  in  3  days,  that  6  men  can  dig  in  4  days  ? 

31.  Suppose  2  men  start  from  the  same  place,  and 
travel  in  opposite  directions,  one  at  the  rate  of  5  miles 
an  hour  and  the  other  f  as  fast; —  how  far  apart  will  they 
be  in  11  hours  ? 

32.  A  fox  has  35  rods  the  start  of  a  greyhound,  but 
the  hound  runs  10  rods  while  the  fox  runs  7.     How 

•  many  rods  must  the  hound  run  to  catch  the  fox }    ^ 

33.  A  started  on  a  journey,  and  travelled  5  miles  an 
hour — B  started  on  the  same  journey,  2  hours  after, 
and  travelled  7^  miles  an  hour.  In  how  many  hours 
did  B  overtake  A  ? 

34.  A  jockey  paid  9  times  as  much  for  his  horse  as 
he  did  for  his  saddle;  he  paid  3  times  as  much  for  his 
saddle  as  he  did  for  his  bridle;  and  for  his  bridle  he  paid 
5  dollars.     What  did  the  whole  cost? 

35.  Suppose  a  man  can  rea}»  ^  of  a  field  of  wheat  in 
a  day,  and  his  son  can  reap  ^  cS  it  in  a  day; — what  part 
of  it  can  they  both  reap  in  a  day  ?  In  what  time  can 
they  both  reap  the  whole  ? 

36.  A  boy  being  asked  how  much  money  he  had, 
replied — '  If  I  had  as  much  more,  and  ^  as  much  more, 
and  I  as  much  more  as  I  really  have,  I  should  then  have 
70  cents.'     How  much  must  he  have  had  ? 

37.  A  gentleman  paid  85  dollars  for  5  weeks'  board 
of  himself,  his  son,  and  one  servant,  at  a  hotel — His 
own  board  cost  twice  as  much  as  his  son's,  and  his  son's 
cost  three  times  as  much  as  the  servant's.  What  was 
the  expense  of  each,  per  week } 


88 


NOTE  TO  TEACHERS. 

The  teacher  should  i>ow  be  provider!  with  "  A  KEY  TO  THE  NORTH 
AMERICAN  ARITHMETIC,"  otliervvise  he  must  lose  much  time  in  examin- 
ing operations.  Tlie  Key  is  a  small  book  designed  exclusively  for  teachers, 
and  contains  answers  to  all  the  examples  in  the  Written  Arithmetic.  If  the 
Key  cannot  be  obtained  at  every  place  where  the  Arithmetic  is  for  sale,  it 
may  still  be  obtained  from  the  publisliers  of  the  Arithmetic,  and  from  the 
principal  book-stores  in  the  larger  cities  and  towns. 

A  variety  of  expedient  methods  may  be  pursued,  in  examining  written  opera- 
tions in  arithmetic;  and  perhaps  no  one  system  can  be  adopted,  from  which  it 
will  not  be  found  advantageous,  occasionally,  to  depart.  My  own  practice  for 
several  years,  with  occasional  variation,  has  been  as  follows. 

A  certain  number  of  examples  having  been  assigned  for  a  lesson  the  day  pre- 
vious, each  scholar  is  supposed  to  be  })repared  with  the  solutions  upon  his  slate, 
and  the  class  are  paraded  for  recitation.  Every  scholar  passes  his  slate  into  the 
hands  of  the  scholar  next  above  him,  except  the  head  scholar,  who  hands  his  to 
tlie  foot  scholar.  The  first  scholar  then  reads  from  the  slate  he  holds,  the  answer 
to  the  first  example;  and  the  teacher,  holding  the  key,  declares  the  answer  to 
be  right,  or  wrong.  When  the  answer  has  been  pronounced  right,  it  is  the 
duty  of  every  scholar  who  finds  a  different  answer  upon  the  slate  he  holds,  to 
signify  it,  and  the  error  is  noted  against  the  owner  of  the  slate.  The  first  ex- 
ample being  disposed  of,  the  answer  to  the  second  example  is  read  by  the  second 
scholar,  and  disposed  of  in  like  manner.  Thus  the  readino^  of  answers  goes 
through  the  class,  and  each  scholar  detects  the  errors  of  his  neighbour.  Individ- 
ual scliolars  are  occasionally  called  upon  to  explain  their  work  in  a  particular 
example,  and  to  give  their  reasons  for  the  operation  adopted.  By  this  mode  of 
examination,  the  work  of  a  large  class  is  particularly  inspected,  in  nearly  the 
same  time  that  would  be  required  to  inspect  the  work  of  one  scholar.  Be- 
sides the  advantage  of  despatch  in  this  mode  of  examination,  the  exercise  it- 
self is  beneficial  to  the  pupils. —  Each  scholar  acts  the  part  of  an  inspector — he 
is  interested  to  be  critical — he  acquires  a  facility  in  deciphering  the  work  of 
others — and  thus  his  perceptive  powers  are  cultivated,  and  a  habit  of  alertness 
IS  attained. 

Before  the  learners  attempt  to  perform  operations  by  figures,  they  should  be 
able  to  write  figures  with  facility,  and  to  arrange  them  regularly.  To  attain 
this  object,  the  arrangement  of  figures  below,  may  be  repeatedly  copied  upon 
the  slate,  until  a  good  degree  of  despatch  and  accuracy  is  acquired. 


^2  3Jj-5  6  ^23A5  6  ^  2  3  Jj- 5  6 

7890^2  7890^2  7890^2 

3j^5  6  7  8  3^5  6  7  8  3^5  6  7  8 

9  0^23^  90^23J^  9  0^23Ja 

567890  567890  567890 

^23Ji-56  ^23A56  ^  2  3  J^  5  6 

7890/2  7890/2  7890/2 

3^5  67  8  3A5  6  78  3^5  67  8 


80 


WRITTEN   ARITHMETIC. 

CHAPTER   I. 
]VUMERATIO]V. 

Section    1. 

The  unit,  which  is  the  first  thing  to  be  considered  in 
numeration,  signifies  One.  The  figure  1  stands  for  one 
unit;  2,  for  two  units;  3,  for  three  units;  4,  for  four 
units;  5,  for  five  units;  6,  for  six  units;  7,  for  seven 
units;   8,  for  eight  units;    9,  for  nine  units. 

The  TEN  is  a  number  which  is  made  up  of  ten  units. 
One  ten  is  expressed  thus,  10;  two  tens,  thus,  20;  three 
tens,  thus,  30;  four  tens,  thus,  40;  &c. 

The  HUNDRED  is  a  number  which  is  made  up  of  ten 
tens.  One  hundred  is  expressed  thus,  100;  two  hun- 
dreds, thus,  200;  three  hundreds,  thus,  300;  &c. 

Suppose  the  balls  below,  which  are  arranged  in  three 
places,  to  represent  8  units,  3  tens,  and  1  hundred. 

HUNDRED  TENS  UNITS 

# 
# 

# 

m 
m 

# 


138 
Learn  from  the  figures  above,  that  the  first  or  right 
hand  figure  expresses  units,  the  second  figure  expresses 
tens,  and  the  third  figure  expresses  hundreds. 

H=^ 


90  WRITTEN   ARITHMETIC.  I. 

The  THOUSAND  is  a  number,  which  is  made  up  of  ten 
hundreds.  One  thousand  is  expressed  thus,  1000;  two 
thousand,  thus,  2000;  three  thousand,  thus,  3000;  &c. 
Observe,  that  a  figure  expresses  thousands,  when  it 
stands  in  the  fourth  place  from  the  right;  therefore  ten 
thousand  is  expressed  thus,  10000;  and  a  hundred 
thousand,  thus,  100  000. 

Examine  the  following  Mimeration  Table.  Begin  at 
the  right  hand,  and  observe,  that  every  three  figures  may- 
be viewed  by  themselves; — the  first  three  express  so 
many  units ^  tens  and  hundreds;  the  second  three,  so 
many  Thousands;  the  third  three,  so  many  Millions; 
the  fourth  three.  Billions;  the  fifth  three,  Trillions.* 


.2^  £.§     ^.2,.^  la 


-^£2  ^'-sg  ^sg  ^■B<  ^ 

Cr;^      CrJi-J       CrjM      5c^      »^Cn^ 

HJ^E-  ajE-M  SE^S  KHH  KE^;^ 
472  156  795  841  526 

To  read  the  line  of  figures  in  this  table,  begin  with 
the  left  hand  figure,  and  proceed  as  follows. 

-§         Ij  S-^^  §1  11 

.^  c  ^  ^   ,  cS    s  -^^  *>  ►L^  -^  fi 

tJ^aa^     C-vJ^M     wC«C     cui:2oy3-M3 

472  156  795  841  526 
This  character,  0,  called  nought^  or  cipher^  expresses 
nothing  of  itself —  it  stands  only  to  occupy  a  place, 
where  there  is  none  of  the  denomination  belonging  to 
that  place  to  be  expressed.  For  example,  in  the  num- 
ber 240,  there  are  no  units;  therefore  a  cipher  stands  in 
the  units'  place.  In  the  number  407,  there  are  no  tens; 
therefore  a  cipher  stands  in  the  tens'  place. 

♦  The  old  method  of  embracing  six  figures  in  a  period,  is  of  late  abandoned  ' 


8. 


NUMERATION. 


91 


Note  to  Teachers.  Require  the  learners  to  copy  upon  their  slates  tlie  fol- 
lowing: ligures  expressing  niunbers.  Then  require  tliem  to  read  from  their  siutes 
tlie  severe,  numbers  expressed. 


(Ex.  1.) 

508 

(19) 

1  000  001 

(2) 

3  861 

(20) 

90  040 

(3) 

1050 

(21) 

107090 

(4) 

27  400 

(22) 

6  000  304 

(5) 

13  008 

(23) 

77  010  000 

(6) 

29111 

(24) 

100100  011 

(7) 

112  600 

(25) 

220  002 

(8) 

30  030 

(26) 

11333111 

(») 

206  209 

(27) 

216  090  900 

(10) 

500  OSS 

(28) 

10  000004 

(11) 

7  432  040 

(29) 

8  000  000  500 

(12) 

200  005 

(30) 

50  000  000  036 

(13) 

9  070  638 

(31) 

1  000  700  007 

(14) 

3  018  103 

(32) 

8  400  052  000  600 

(15) 

16  974  036 

(33) 

8  631008  000 

(16) 

340  007  140 

(34) 

22  000  004 

(17) 

31  031  032 

(35) 

919  000  000  060 

(18)  . 

9  908  000 

(36) 

86  000  001100  018 

Section   2. 

Note  to  Teachers.  The  following  numbers  written  in  words,  are  to  be 
written  upon  the  slate  in  figures.  If  the  learner  meet  with  difficulty  in  denoting 
the  larger  numbers,  he  may  be  instructed  to  repeat  the  Numeration  Table, 
from  units  up  to  the  highest  denomination  in  the  number  to  be  denoted;  and, 
while  repeating  the  table,  he  may  make  a  dot  for  each  denomination,  arranging 
the  whole  in  a  line.  Then,  the  figure  to  express  the  highest  denomination  may 
be  written  under  the  left  hand  dot,  and  there  will  be  no  difficulty  in  arranging 
the  figures  of  other  denominations  under  their  respective  dots. 

1.  Seventy. 

2.  Forty-eight. 

3.  One  hundred  and  twenty-four. 

4.  Six  hundred  and  nine. 

5.  Three  thousand,  and  six  hundred. 

6.  Two  thousand,  four  hundred  and  fifty. 

7.  Nineteen  thousand,  and  sixty-eight. 

8.  Five  thousand,  seven  hundred  and  tbirty-one. 


99  WRITTEN  ARITHMETIC.  I 

9.  Thirty-six  thousand,  seven  hundred  and  forty. 

10.  Two  hundred  and  sixty-eight  thousand. 

11.  Nine  hundred  five  thousand,  and  one  hundred. 

12.  Eighteen  thousand,  seven  hundred  and  thirty-five 

13.  Seven  hundred  thousand  and  nine. 

14.  Thirteen  million,  sixteen  thousand,  and  nineteen. 

15.  One  hundred  five  million,  two  thousand,  and  one. 

16.  Six  billion,  forty  million,  and  six  thousand. 

17.  Twenty-one  bilHon,  and  one  hundred  million. 

18.  Five    trillion,    fourteen  billion,  seventy  million, 
one  thousand,  two  hundred  and  thirty-six. 

19.  One  hundred  twenty-two  trillion,  eight  hundred 
and  forty-seven  thousand. 

20.  Ten  bilhon,  nine  hundred  eighty-seven  thousand, 
seven  hundred  and  thirty. 

21.  Seven  hundred  trillion,  and  thirty-six  thousand. 

22.  Twelve  billion,  eight  hundred  forty-two  thousand, 
seven  hundred  and  eighty. 

23.  Twenty-nine  trillion,  eight  hundred  nine  billion, 
one  thousand,  and  eighteen. 

24.  Eight  hundred  twenty-three  billion,  ten  million, 
eight  thousand,  and  fifteen. 


Questions  to  be  answered  Orally, 
(1)  What  is  a  unit?  (2)  What  is  the  greatest 
number,  that  can  be  expressed  by  one  figure  alone  ? 
(3)  In  what  situation  must  the  figure  9  stand,  to 
express  9  tens  ?  (4)  What  is  the  greatest  number 
that  can  be  expressed  by  two  figures  ?  (5)  Recite 
the  several  denominations  of  numbers,  from  units  to 
trillions  J  as  they  stand  in  the  Numeration  Table. 
(6)  What  denominations  are  expressed  in  the  1st. 
three  places  of  figures  ?  (7)  What  denominations 
are  expressed  in  the  2nd.  three  places  ?  (8)  Where 
must  the  figure  7  stand  to  express  7  tens  of  thousands 
—  that  is,  seventy  thousand  ?  (9)  What  denomina- 
tions are  expressed  in  the  3rd.  three  places  ? 
(10)  Where  must  the  figure  2  stand,  to  express  two 
hundred  thousand  ? 


1.  2.  ADDITION  M 

CHAP.   II. 
ADDITIOIV. 

Section  1. 
1.  What  is  the  whole  sum  of  5312  dollars,  8032  doK 
lars,  601  dollars,  and  7123  dollars  ? 

w  ^  We  first  write  the  nunibers  under  one 

g  *g  another,  so  that  all  the  units  may  stand  in  a 

I'l  «^        column  on  the  right  hand.     We  then  add 

6  3  12       ^^^  six,  and  2  are  eight;  and  we  write  8 
8  0  3  2       under  the  column  of  units.     We  next  add 

5  0  1  the  column  of  tens,  and,  finding  their  sum 

7  12  3  to  be  6,  we  write  6  under  the  column.  In 
■  -  ck  i^  Q  ^^®  same  manner  we  add  the  hundreds,  and 
^^  -^^^  the  thousands. 

Find  the  sum  of  the  numbers  in  each  of  the  following 
examples,  by  addition  upon  the  slate. 

(2).  51  (3).  733  (4).  6243  (5).  24031 

4  120  4123  1320 

60  12  9401  40214 

43  634  130  34314 


Section  2. 
1.  Add  the  following  numbers  into  one  sum.     4638 
and  216  and  8329  and  1212. 

^  ^  Finding  the  sum  of  the  units  to  be  25,  or 

§1  2  tens  and  5  units,  we  write  only  the  5  units, 

I"?  gS        and  presently  add  the  2  tens  in  with  the 

hE^5       column  of  tens.     In  adding  the  hundreds, 

4  6  38       we  find  their  sum  to  be  13.     Now  if  we 

216       should  write  down  13,  the  3  would  stand 

8  32  9       under  the  column  of  hundreds,  and  the  1, 

12  12        under  the  column  of  thousands;  therefore 

14  3  9  5       we  write  the  3  only,  and  presently  add  the 

• 1  in  with  the  thousands. 


94  WRITTEN     ARITHMETIC.  H 

In  the  following  examples,  observe,  that  when  the 
sum  of  any  column  amounts  to  more  than  9,  you  must 
set  down  only  the  right  hand  figure  of  it,  and,  must  add 
the  left  hand  figure  to  the  next  column. 

(2).  6214  (3).  5221  (4).  7420  (5).  3150 

2403  7540  612  216 

590  1368  2541  8481 

8732  520  9103  275 

1217  5648  430  8610 

2464  ^7300  1000  2541 


RULE  FOR  ADDITION.  W^'ite  the  numbers^  units  iindet 
unitSj  tens  under  tens^  ^c,  Md  each  column  separately^ 
beginning  with  the  column  of  units.  When  the  sum  of 
any  column  is  not  more  than  9,  lorite  it  under  the  column: 
when  the  sum  is  more  than  9,  write  only  the  units^  figure 
under  the  column^  and  carry  the  number  of  tens  to  the 
next  column.  Finally^  write  down  the  whole  sum  of  the 
left  hand  column. 

6.  Add  together  the  numbers,  143  and  8  and  56  and  7. 

7.  Add  together  the  numbers,  3  and  96  and  5  and  984. 

8.  What  is  the  whole  sum  of  26,  9,  18,  153  and  728  ? 

9.  What  is  the  whole  sum  of  8,  6,  42,  728  and  4105? 

10.  What  is  the  whole  sum  of  44,  536,  827  and  3480  ? 

11.  What  is  the  whole  sum  of  1118,  6004,  and  84932  ? 

12.  What  is  the  whole  sum  of  61297,  58  and  389166  > 

13.  Find  the  sum  of  423,  315,  531,  414,  612,  234, 
621,  414,  711,  144,  621  and  918. 

14.  Find  the  sum  of  314,  90,  246518,  7,  1101,  47, 
3430,  8601520,  2004  and  5674. 

15.  Find  the  sum  of  1728,  26510,  34,  100,  3261,  9, 
245,  1640831,  6733  and  40000000. 

16.  A  clerk  received  from  one  man  94  dollars,  from 
another  361  dollars,  and  from  another  113  dollars. 
What  was  the  v?4iole  sum  of  moQey  received  ^ 

17.  A  merchant  sent  to  the  bank  at  one  time  301  dol- 
lars; at  another  214;  at  another  1109;  at  another  109 
How  much  did  he  send  in  all  ? 


U  ADDITION. 


vim 


IS.  A  certain  lot  of  land  has  been  divided  into  three 
farms;  one  of  the  farms  contains  112  acres,  another  123 
acres,  and  the  other  147  acres.  How  many  acres  were 
there  in  the  original  lot  ? 

19.  If  you  start  on  a  journey,  and  travel  on  Monday 
42  miles,  on  Tuesday  57,  on  Wednesday  49,  on  Thurs- 
day 54,  on  Friday  63,  and  on  Saturday  75,  how  far 
will  you  have  travelled  at  the  end  of  the  week  ? 

20.  Suppose  477  dollars  are  in  one  bag,  8509  in 
another,  1965  in  another,  and  956  in  another;  what  sum 
of  money  is  there  in  the  four  bags  ? 

21.  A  merchant  bought  a  quantity  of  sugar  for  2075 
dollars,  and  then  sold  it  so  as  to  gain  415  dollars.  For 
how  much  did  he  sell  the  sugar  ? 

22.  There  are  four  numbers,  the  first  of  which  is  532, 
the  second  895,  the  third  240,  and  the  fourth  as  much 
as  the  other  three.     What  is  the  sum  of  them  all  ? 

23.  A  broker,  by  selhng  a  note  for  836  dollars,  lost 
140  dollars.     What  must  he  have  paid  for  the  note  ? 

24.  A  capitalist  gave  to  one  of  his  sons,  13427  dol- 
lars; to  another,  13025  dollars;  to  another,  12947  dol- 
lars.    What  did  he  give  to  all  of  them  ? 

25.  Sacred  history  shows,  that  the  time,  from  the 
creation  of  the  world  to  the  Deluge,  was  1656  years; 
thence  to  the  building  of  Solomon's  temple,  1344  years; 
thence  to  the  birth  of  Christ,  1004  years.  How  old  is 
the  world  the  present  year  ? 

26.  George  Washington  was  born  in  the  year  1732, 
and  lived  to  be  67  years  old.     In  what  year  did  he  die  } 

,27.  Three  men  united  in  trade; — the  first  man  had 
5136  dollars,  the  second  had  1562  dollars,  and  the  third 
had  756  dollars.     How  much  had  they  all  ? 

28.  A  ti-ader  bought  four  pieces  of  cloth:  the  first 
piece  contained  86  yards;  the  second,  55  yards;  the 
third,  87  yards  and  the  fourth  91  yards.  What  was  the 
cost  of  the  whole,  at  1  dollar  per  yard  ? 

20.  A  gentleman  pur(;hasecl  a  farm  fo^^  8257  dollars; 
paid  953  dollars  for  having  it  fenced,  and  300  dollars 
for  having  a  barn  built  upon  it.  For  how  much  must 
he  sell  it,  in  order  to  2;ain  100  dollars  ? 


96  WRITTEN   AKlTHxMETlC.  ]J 

30.  A  dfover  paid  300  dollars  for  100  sheep,  525 
dollars  for  150  sheep,  and  1000  dollars  for  250  sheep. 
How  many  did  he  buy  ?     What  did  the  whole  cost  ? 

31.  What  is  the  sum  of  two  million,  five  hundred 
thirty-one  thousand,  one  hundred  and  twenty, — -fourteen 
thousand, — thirty  thousand  and  twenty-four, — five  hundred 
und  sixty, — and  seven  hundred  and  two  ? 

32.  The  inhabitants  of  the  British  Islands  are  stated 
thus:  England,  11260  555;  Wales,  717  103;  Scotland, 
2092014;  Ireland,  6  846  949  ;  Army  and  Navy,  310  000; 
Isle  of  Man,  40  981 ;  Guernsey,  20  827  ;  Jersey,  28  600 ; 
Scilly  Isles,  2  614.     What  is  the  whole  number? 

33.  The  inhabitants  of  the  United  States,  by  the  census 
of  1840,  were  stated  thus:  Maine,  501  793;  New  Hamp- 
shire, 284  574;  Massachusetts,  737  699;  Rhode  Island, 
108  830  ;  Connecticut,  309  978 ;  Vermont,  291  948 ;  New 
York,  2  428  921;  New  Jersey,  373  306;  Pennsylvania, 
1  724  033  ;  Delaware,  78  085  ;  Maryland,  470  019;  Vir- 
ginia, 1  239  797;  North  Carolina, 753  419;  South  Carolina, 
594  398 ;  Georo;ia,  691  392 ;  Alabama,  590756 ;  Mississippi, 
375  051;  Louisiana,  352  411;  Tennessee,  829  210;  Ken- 
tucky, 779  828;  Ohio,  1  519  467  ;  Indiana,  685  866;  Illi- 
nois," 476  183 ;  Missouri,  383  702 ;  Arkansas,  97  574  ;  Mi- 
chigan, 212  267;  Florida,  54  477;  Wiskonsan,  30  945; 
Iowa,  43  112 ;  District  of  Columbia,  43  712.  What  was 
the  whole  number  ? 


Questions  to  he  ansi/mred  Oralhj, 

(1)  When  you  have  several  numbers  to  add  to- 
gether, in  what  order  do  you  write  them  ?  (2)  Which 
column  do  you  add  first  ?  (3)  Do  you  add  all  other 
columns  in  tbe  same  manner  that  you  add  the  first  ? 
(4)  When  the  sum  of  any  column  is  less  than  10, 
where  is  it  to  be  written?  (5)  When  the  sum  of 
any  column  is  more  than  9,  what  is  to  be  done? 
(6)  Why  do  we  carry  as  many  ones  to  the  next  left 
hand  column,  as  there  are  tens  in  any  column  that  we 
have  added  ?     (7)    Recite  the  rule  for  addition. 


I.   2.  SUBTRACTION.  97 

CHAP.  III. 
SUBTRACTIOIV, 

Section  1. 

1.   Subtract  632  from   1847;  that  is,  take  632  from 

1347,  and  find  what  number  remains. 

1847  ^^  ^^^^^  write  the  smaller  number  under 

^32   "    the  greater.     Then,  take  2  units  from  7  units, 

^  ^  -       3  tens  from  4  tens,  6  hundreds  from  8  hun- 

.__J — L       dred,  and  nothing  from  1  thousand. 

Subtract  the  smaller  number  from  the  greater  in  each 
of  the  folio w^ing  examples. 

(2).  25    (3).  639    (4).  4258    (5).  705684 
12       213       3215         4261 


6.  A  farmer  having  359  sheep,  sold  136  of  them,  and 
kept  the  remainder.     How  many  did  he  keep  ? 

7.  A  trader  having  2748  dollars,  laid  out  2616  dollars 
for  goods.     How  many  dollars  had  he  remaining  ? 

Section   2. 
1.   Subtract  the  number  1528  from  the  number  8473. 
We  unite  1  of  the  7  tens  with  the  3  units, 
847  3       rnaking  13  units,  and  say,  8  from  13,  leaves 
.?_?_?.       5.     Then,  having  used  1  of  the  7  tens,  w^e 
694  5       take  2  tens  from  6  tens.     In  the  same  way 
■       we  take  5  hundreds  from  14  hundreds. 
Do  not  pass  from  the  abov^  example  without  under- 
standing it.     Whenever  an  upper  figure  is  smaller  than 
the  figure  under  it,  we  use  1  from  the  next  upper  figure, 
and  this  1  becomes  10  w^hen  considered  with  the  right 
hand  figure.     Arithmeticians  commonly  call  this  process, 
borrowing  10;  and,  instead  of  reckoning  the  figure  from 
which  they  have  borrowed  to  be  1  less  than  it  stands, 
they  pay  1  to  the  figure  under  it — reckoning  the  lower 
figure  to  be  1  morp*  than  it  stands. 

I 


98  W.RITTEN    ARITHMETIC.  IL. 

Perform  subtraction  in  the  following  examples. 

(2).  1853   (3).  5264   (4).  2657   (5).  6807 
1370        762        349      4096 


6.   Subtract  1268  from  1503. 

In  subtracting  the  8  units,  we  use  a  ten^ 

15  03       ^i^r^^  y^Q  obtain  by  supposing  1  of  the  5  hun- 

^^^^       dreds,  (which  is  10  tens,)  to  be  where  the 

235       0  is.     Then,  having  used  1  of  the  10  tenSj 

we  presently  subtract  6  tens  from  9  tens 

7    Subtract  1146  from  2601. 

8.  Subtract  5428  from  8019. 

9.  Subtract  258  from  34307. 

RULE  FOR  SUBTRACTION.  Write  the  smaller  number 
under  the  greater^  placing  units  under  units^  ^c.  Begin 
with  the  units^  and  subtract  each  figure  in  the  lower 
number  from  the  figure  over  it.  When  a  figure  in  the 
upper  number  is  smaller  than  the  figure  under  it^  consider 
the  upper  figure  to  be  10  more  than  it  is^  and  the  next 
upper  figure  on  the  left  hand^  to  be  1  less  than  it  is. 

PROOF,  '^dd  together  the  remainder  and  the  smaller 
number :  their  stun  will  be  equal  to  the  greater  number ^ 
if  the  work  be  right. 

10.  Find  the  difference  between  39  and  64,  by  sub- 
tracting the  smaller  number  from  the  greater. 

11.  What  is  the  difference  between  464  and  502  ? 

12.  What  is  the  difference  between  99  and  200  ? 

13.  What  is  the  difference  between  35720  and  9100  ? 

14.  Subtract  44  from  10000. 

15.  I  deposited  1450  dollars  in  the  bank,  and  I  have 
since  drawn  out  835  dollars.  How  many  dollars  have 
I  remaining  in  the  bank  ? 

16.  Suppose  a  man  owes  1634  dollars,  and  possesses 
property  to  the  amount  of  8150  dollars;  how  much  will 
he  have  left,  after  paying  his  debts  ? 

17.  Subtract  sixty-two  thousand  five  hundred  and 
seven,  from  one  million  eighty  thousand  and  forty-four. 


O^  SUBTRACTION.  99 

IS.  The  number  of  inhabitants  in  the  city  of  London 
is  1  250  000;  the  number  in  the  city  of  Paris  is  750  000 
How  many  more  are  there  in  London,  than  in  Paris  .^ 

19.  The  population  of  Great  Britain  and  Ireland  is 
21  500  000;  the  population  of  France  is  32  000  000. 
How  many  more  inhabitants  are  there  in  France,  than 
in  Great  Britain  and  Ireland? 

20.  The  Rocky  Mountains,  in  North  America,  rise 
12  500  feet  above  the  level  of  the  ocean;  the  Andes,  in 
South  America,  rise  21440  feet.  How  many  feet 
higher  are  the  latter,  than  the  former? 

21.  A  merchant  paid  13  745  dollars  for  a  ship,  and 
sold  it  for  15  150  dollars.     What  did  he  gain? 

22.  A  farmer  sold  a  piece  of  wood-land  for  396  dol- 
lars, which  was  78  dollars  more  than  he  gave  for  it. 
How  much  did  he  give  for  the  land? 

23.  Columbus  discovered  America  in  the  year  1492. 
How  many  years  is  it  since  the  discovery? 

24.  The  United  States  declared  Independence  in  the 
year  1776.     How  many  years  since  the  declaration? 

25.  A  man  bought  20  casks  of  wine,  containing  2459 
gallons,  and  sold  14  casks  containing  1682  gallons. 
How  many  casks,  and  how  many  gallons  were  left? 

26.  There  are  two  numbers,  whose  difference  is  758; 
the  greater  number  is  1524.     What  is  the  smaller  number? 


Q^uestions  to  be  answered  Orally, 
(1)  How  can  you  find  what  the  difference  is  be- 
tween two  numbers?  (2)  When  one  number  is  to 
be  subtracted  from  anothegjttfcvhat  order  must  the 
numbers  be  written?  (3)^[ff  what  place  do  you 
begin  to  perform  the  subtraction?  (4)  When  a 
figure  in  the  upper  number  is  smaller  than  the  figure 
under  it,  what  is  to  be  done?  (5)  Where  does  the 
remainder  appear,  after  the  subtraction  is  performed? 
(6)  Recite  the  rule  for  subtraction.  (7)  How  can 
you  prove  that  an  operation  in  subtraction  is  per- 
formed correctly? 


100  WRITTEN   ARITHMETIC.  I. 

Section  3. 

MISCELLANEOUS   EXAMPLES. 

1.  A  man  owing  379  dollars,  paid  at  one  time  47 
dollars,  at  another  23,  at  another  84,  and  at  another,  143. 
How  much  did  he  still  owe? 

2.  There   are  1000  dollars  in  4  bags;    the  first  bag 
contains  230  dollars,  the  second  245,  the  third  270 
What  is  contained  in  the  fourth  bag? 

3.  Suppose  the  world  to  have  been  created  4004 
years  before  the  Christian  era,  how  old  is  it  at  this  date? 

4.  A  man  having  in  his  desk  2000  dollars,  took  out 
120  dollars  to  pay  a  debt,  and  afterwards  put  in  75  dols. 
How  much  w^as  there  remaining  in  the  desk? 

5.  A  merchant  bought  a  ship  for  11  240  dollars,  paid 
305  dols.  for  repairing  it,  and  sold  it  so  that  he  lost  95 
dols.     For  how  much  did  he  sell  it? 

6.  What  is  the  sum  of  58,  45,  and  70?  Then,  if  you 
subtract  43  from  this  sum,  what  will  be  the  remainder? 

7.  A  merchant,  who  had  650  barrels  of  flour,  sold  95 
barrels  to  one  man,  38  to  another,  and  225  to  another. 
How  many  barrels  had  he  left? 

8.  A  jockey  bought  a  horse  for  115  dollars;  he  ex- 
changed him  for  a  better  horse,  paying  23  dollars,  and 
then  sold  the  better  one  for  137  dollars.  Did  he  gain 
or  lose? — and  how  much? 

9.  If  654  be  subtracted  from  10000,  and  then  29670 
be  added  to  the  remainder,  what  will  be  the  sum? 

10.  A  gentleman  gave  972  dollars  for  a  carriage  and 
two  horses;  the  carriage  was  valued  at  525  dollars. 
What  was  the  value  ofj^j^orses? 

11.  Dr.  Franklin  diWW  the  year  1790,  and  he  was 
84  years  old  when  he  died.     In  what  year  was  he  born? 

12.  A  clerk  went  out  with  240  dollars,  to  settle  some 
accounts:  he  paid  126  dollars  to  one  man,  received  37 
dollars  from  another,  and  paid  94  dollars  to  another. 
How  many  dollars  had  he  then? 

13.  Add  together  two  hundred,  sixteen  thousand, 
thirteen  million,  and  seven  billion;  and  then  subtract  ten 
thousand  from  the  sum 


1.2.  MULTIPLICATION  101 

CHAP.  IV. 
MULTIPLICATIOJV. 

Section  1. 

1.  If  a  gunner  shoot  72  pigeons  every  time  he  goes  a 
gunning,  how  many  will  he  shoot  m  going  3  times  ? 

We  might  here  obtain  the  answer  by  adding  together, 

72  and  72  and  72;  but  we  shall  obtain  it  more  readily 

by  multiplying  72  by  3;  that  is,  by  finding  3  times  72. 

Multiplicand     72  We  write  72,  and  write  3  under 

Multiplier  3        it.     Then  we  multiply  the  2  units 

Product           216        ^."^  '^l  ^  '^f  separately,  thus,  3 
times  2  are  6;  3  times  7  are  21. 

Observe,  that  the  number  which  we  multiply  is  called 
the  multiplicand;  the  number  by  which  we  multiply  is 
called  the  multiplier;  and  the  number  which  we  obtain 
by  multiplication  is  called  the  product. 

Find  the  product  in  each  of  the  following  examples. 

(2).  61         (3).  524        (4).  9132        (5).  420121 
4  2  3  4 


6.  If  a  farm  produce  230  bushels  of  wheat  a  year, 
how  many  bushels  will  it  produce  in  3  years  ? 

7.  Multiply  512  by  4; — that  is,  find  4  times  512. 

Section  2. 
1.  Multiply  743  by  6; — that  is,  find  6  times  743. 
743  6  times  3  are  18,  or  1  ten  and  8  units;  we 

(3       write  only  the  8  units,  (as  in  addition),  and 
'  ,  -  Q        proceed; — 6  times  4  are  24  and  1  we  carry 

are  25;  we  write  the  5  and  proceed. 

Find  the  product  in  each  of  the  following  examples. 

(2).  5236        (3).   1908        (4).  6175        (5).  3640 
4  2  .5  8 


102  WRi^TTEN     ARITHMETIC.  IV. 

6.  What  will  3  books  cost,  at  31  cents  apiece  ? 

7.  What  will  4  slates  cost,  at  24  cents  apiece  ? 

8.  What  will  5  baskets  cost,  at  17  cents  apiece? 

9.  What  will  6  ^ows  cost,  at  25  dollars  apiece  ? 

10.  What  will  7  horses  cost,  at  115  dollars  apiece? 
1 1     How  niany  are  8  times  9  ? 

12.  How  many  are  9  times  16  ? 

13.  How  many  are  5  times  342? 

14.  How  many  are  7  times  6453  ? 

15.  How  many  are  3  times  42  90S  ? 

16.  How  many  are  6  times  704  370?  ^ 

17.  Multiply  251  by  8, — that  is,  find  8  times  251. 

18.  Multiply  475  by  4. 

19.  Multiply  3086  by  6. 

20.  Multiply  15  350  by  8. 

21.  Multiply  430  039  by  7. 

22.  Multiply  7  000  005  by  9. 

23.  Multiply  42  862  000  by  5. 

24.  Multiply  928  064  309  by  4. 

25.  Suppose  8  to  be  a  multiplicand,  and  6  the  multi 
plier;  how  much  will  be  the  product? 

26.  Suppose  35  to  be  a  multiplicand,  and  7  the  multi 
plier;  how  much  will  be  the  product? 

27.  Suppose  491  to  be  a  multiplicand,  and  5  the  mul- 
tipher;  how  much  will  be  the  product? 

Section   3. 

1.  Multiply  657  by  24. 

We  first  multiply  by  the  4  units.     Then 
^  we  multiply  by  the  2  tens,  and  since  this 

ZZ        product  must  be  ten  times  greater  than  it 

2628  would  be  if  the  2  were  2  units,  we  set  the 
1314  product  one  place  to  the  left.  At  last,  we 
15768  ^^^  ^^^  ^^^'^  products  together,  and  the  sura 
•        is  the  whole  product  of  657  by  24. 

2.  Multiply  75  by  16. 

3.  Multiply  6?4  by  45.. 

4.  Multiply  3291  bv  63. 
6.  Multiply  71  538  by ''7. 
6.  Multiply  428  601  by  81. 


3.  MULTIPLICATION.  103 

RULE  FOR  MULTIPLICATION.  Write  the  multiplier  un- 
der the  multiplicjnd^  placing  units  under  units^  <^c. 

When  there  is  but  one  figure  in  the  multiplier^  begin 
with  the  units  J  multiply  each  figure  in  the  multiplicand 
separately^  and  place  each  product  under  the  figure  in  the 
multiplicand  from  which  it  arose;  observing  to  carry  the 
tens  to  the  left  as  in  addition. 

When  there  is  more  than  one  figure  in  the  multiplier j 
multiply  by  each  figure  separately^  and  tcrite  its  product 
in  a  separate  line^  placing  the  right  hand  figure  of  each 
line  under  the  figure  by  which  you  multiply;  and  final" 
ly^  add  together  the  several  products.  The  sum  will  be 
the  whole  product. 

7.   Suppose  5  476  208  to  be  a  multiplicand,  and  3942 
the  multiplier;  how  much  will  be  the  product.'^ 
5476208 
3942 


10952416 
21904832 

49285872 
16428624 

21587211936 

8.  Suppose  73  054  to  be  a  multiplicand,  and  548  th.'' 
multiplier ij how  much  will  be  the  product.^ 

9.  Suppose  295  to  be  a  multipHcand,  and  486  the 
multiplier;  what  will  be  the  product } 

10.  What  is  the  product  of  9351  by  765  ? 

11.  What  is  the  product  of  3008  by  254  ? 

12.  What  is  the  product  of  5603  by  6448  ? 

13.  How  many  are  74  times  6580  ? 

14.  How  many  are  236  times  3759  ? 

15.  There  is  an  orchard  containing  9  rows  of  trees, 
and  there  are  57  trees  in  each  row.  How  many  trees 
are  there  in  the  orchard  ? 

16.  A  merchant  bought  75  pipes  of  wine,  at  145  dol- 
lars a  pige.     What  did  the  whole  cost } 

17.  A  merchant  bought  37  mules,  for  shipping,  at  52 
dollars  per  head.     What  did  the  whole  cost } 


104  WRITTEN   ARITHMETIC.  IV 

18.  A  man  travelled  26  days,  travelling  47  miles  a 
flay.     How  far  did  he  travel  in  the  whole  time? 

19.  A  merchant  sold  342  tons  of  iron,  at  142  dollars 
per  ton.     What  was  the  price  of  the  whole? 

20.  If  a  coach  wheel  turn  round  346  times  in  1  mile, 
how  many  times  will  it  turn  round  in  the  distance  from 
New  York  to  Philadelphia,  it  being  95  miles? 

21.  A  prize  was  divided  among  47  men,  and  each 
man  received  25  dollars.     How  much  was  the  prize? 

22.  What  sum  of  money  must  be  divided  among  4*^ 
men,  so  that  each  man  shall  receive  59  dollars? 

23.  A  merchant  bought  7  bales  of  cloth,  each  balu 
containing  11  pieces,  and  each  piece,  29  yards.  How 
many  pieces,  and  how  many  yards  were  there? 

24.  A  trader  bought  9  pieces  of  cloth,  each  piece 
containing  42  yards,  at  6  dollars  a  yard.  How  many 
yards  were  there,  and  what  did  the  whole  cost? 

25.  If  hats  are  worth  .7  dollars  apiece,  w^hat  are  15 
boxes  of  hats  worth,  each  box  containing  24  hats? 

26.  The  distance  from  Washington  to  Boston  is  436 
miles;  and  in  each  mile  there  are  320  rods.  How 
many  rods  is  it  from  Washington  to  Boston? 

27.  The  distance  from  Washington  to  New-Orleans 
is  1255  miles.     How  many  rods  is  it? 

28.  What  is  the  value  of  the  hay,  that  is  produced  on 
16  farms;  allowing  each  farm  to  produce  62  tons,  and 
allowing  the  hay  to  be  worth  12  dollars  a  toU? 

29.  There  are  24  hours  in  a  day,  and  365  days  in  a 
year.  If  a  ship  sail  7  miles  in  an  hour,  how  many  miles 
will  she  sail  in  a  year? 

30.  How  many  days'  work  can  9  men  do  in  24  days? 

31.  How  many  days  will  it  take  1  man  to  perform  a 
piece  of  work,  that  9  men  will  perform  in  24  days? 

32.  How  many  days  will  it  take  1  man  to  build  a 
piece  of  road,  that  13  men  can  build  in  47  days? 

33.  How  many  men  must  be  employed  to  do  a  piece 
of  work  m  1  day,  that  11  men  can  perform  in  18  days? 

34.  Suppose  that  a  ship's  crew  of  13  men  will  drink 
82  gallons  of  water  in  14  days,  how  long  \fould  the 
same  quantity  of  water  last  1  man? 


4.  MULTIPLICATION  IQB 

Section  4. 

ABBREVIATIONS. 

When  there  are  ciphers  standing  between  figures^  in 
^he  multiplier^  they  may  be  disregarded, 

1.  What  is  the  product  of  12318  12318 

multiplied  by  7004  ?  7004 

49272 
86226 


86275272 


2.  What  is  the  product  of  9651  multiplied  by  304? 

3.  How  many  are  1001  times  57  906  ? 

4.  How  many  are  905  times  820  437  ? 

Ciphers  on  the  right  hand  of  the  multiplier  or  multi" 
plicand  may  be  disregarded  till  the  multiplication  is  per- 
formedy  and  then  placed  on  the  right  hand  of  the  product* 

5.  What  is  the  product  of  5763  5763 
multiplied  by  3600  ?  3600 

34578 
17289 


20746800 


6.  What  is  the  product  of  158  multiplied  by  350  ? 

7.  How  many  are  800  times  369  ? 

8.  How  many  are  40  times  4728  ? 

Ciphers  on  the  right  hand  of  the  multiplier  and  muU 
tiplicand  both^  may  all  be  disregarded  in  multiplying^ 
and  finally  placed  on  the  right  hand  of  the  product* 

9.  What  is  the  product  of  46000  46000 
multiplied  by  340.                                                340 

184 
138 


15640000 


10.  What  is  the  product  of  8370  multiplied  by  240? 

11.  How  many  are  90  times  761000? 

12.  How  many  are  5700  times  6800? 


106  WRITTEN    ARITHMETIC.  lY, 

When  the  multiplier  is  10,  100,  1000,  ^c  merely 
place  the  ciphers  of  the  multiplier  on  the  right  hand  of 
the  multiplicand^  and  it  becomes  the  product, 

13.  What  is  the  product  of  5  multiplied  by  10  ? 

14.  What  is  the  product  of  17  multiplied  by  100.^ 

15.  What  is  the  product  of  49  multiplied  by  1000.^ 

16.  In  1  dollar  there  are  100  cents.  How  many 
cents  are  there  in  6  dollars  ^ 

17.  How  many  cents  are  there  in  25  dollars  ? 

18.  If  1  box  of  lemons  cost  7  dollars,  how  many 
cents  will  it  take  to  pay  for  10  boxes  ? 

When  the  multiplier  is  a  number^  that  can  be  produc- 
ed by  multiplying  two  smaller  numbers  together^  muU 
tiply  the  multiplicand  first  by  one  of  the  smaller  numbers j 
and  the  product  thence  arising  by  the  other. 

19.  Find  the  price  of  32  horses,  at  96  dollars  apiece. 

96  price  of  1  horse. 

8 


768  price  of  8  horses. 
4 


3072  price  of  4  times  8  horses,  or  32  horses. 

Observe  in  the  above  example,  that  32  can  be  pro- 
duced by  multiplying  4  and  8  together.  The  4  and  the 
8  are  called  the  factors  of  32. 

20.  A  merchant  bought  24  hogsheads  of  molasses  at 
19  dollars  a  hogshead.     What  did  the  whole  cost  ? 

In  this  example  we  consider  24  to  be  the  multiplier. 
For  24,  we  can  find  several  different  sets  of  factors; 
viz.  3,  8;  alsQ,  4,  6;  also,  2,  3,  4;  also,  2,  2,  6. 
Either  set  of  these  factors  may  be  used. 

21.  If  a  ship  sail  at  the  rate  of  129  miles  a  day,  how 
many  miles  will  she  sail  in  72  days  ? 

22.  If  1  man  can  dig  41  bushels  of  potatoes  in  a  day, 
now  many  bushels  can  28  men  dig  ? 

23.  Multiply  425  by  36,  using  the  factors  of  36. 

24.  How  many  are  63  times  540  ? 

25.  How  many  are  45  times  2807  7 


MULTIPLICATION. 


\m 


Questions  to  be  answered  Orally, 
(1)  What  is  meant  by  multiplicand? — what  by 
multiplier? — and  what  by  product?  (2)  When  we 
say,  5  times  8  are  40,  which  of  these  numbers  is  the 
muhipHcand  ? — which  the  multipHer  ? — and  which 
the  product  ?  (3)  Con  you  obtain  the  product  of 
any  two  numbers,  by  means  of  addition  ?  (4)  Re- 
cite the  rule  for  multipHcation.  (5)  When  there 
are  ciphers  between  figures  in  the  multiplier,  what 
may  be  done  ?  (6)  When  there  are  ciphers  on  the 
right  of  the  multipher,  or  multiplicand,  or  on  the 
right  of  both,  what  may  be  done  with  them  ?  (7)  In 
what  manner  can  you  multiply  by  10,  by  100,  by 
1000,  &c.?  (8)  What  is  meant  by  the  factors  of  a 
number?  (9)  Name  two  factors  of  24.  (10)  Name 
three  factors  of  24.  (11)  Name  two  factors  of  36. 
(12)  Name  three  factors  of  36. 


Perform  the  following  examples  by  either  of  the  fore- 
going methods,  which  may  be  found  convenient. 

26.  What  is  the  value  of  a  farm  consisting  of  200 
acres  of  land,  at  40  dollars  an  acre  ? 

27.  Suppose  a  book  to  contain  235  pages,  45  lines 
in  each  page,  and  50  letters  in  each  line; — how  many 
letters  are  there  in  the  book  ? 

28.  Suppose  an  orchard  to  consist  of  109  rows,  126 
trees  in  a  row,  and  1007  apples  on  a  tree; — how  many 
trees,  and  how  many  apples  are  there  ? 

29.  Suppose  a  crew  of  fifty  men  have  provision  for  30 
days,  allowing  each  man  20  ounces  a  day; — how  many 
days  would  it  last,  if  each  man  ate  1  ounce  a  day  ? 

30.  Suppose  a  crew  of  fifty  men  have  provision  for 
30  days,  allowing  each  man  20  ounces  a  day; — how 
many  men  would  it  serve  for  the  same  time,  if  each  man 
ate  one  ounce  a  day .'' 

31.  How  many  fishes  would  be  caught  by  14  boats, 
employed  for  30  days,  each  boat  drawing  a  net  15  times 
a  day,  and  taking  13  fishes  each  draught  ? 

32.  What  is  the  product  of  90042  multiplied  by  9009  ? 


108  WRITTEN    ARITHMETIC.  y. 

CHAP.    V. 

DIVlSIOJf. 

Section  1. 

I.  How  many  yards  of  cloth,  at  3  dollars  a  yard,  can 
be  bought  ^or  396  dollars  ? 

Here  we  must  find  how  many  times  3  dollars  there 
are  in  396  dollars:  that  is,  we  must  divide  396  by  3. 

3)39  6  We  first  divide  the  3  hundreds,  then  the 
Y32  ^  tens,  and  then  the  6  units;  thus,  3  in  3, 
~      once;  3  in  9,  3  times;  3  in  6,  2  times. 

Observe  in  the  above  example,  that  the  3  which  we 
first  divide,  means  3  hundred;  and  the  1  which  we  place 
under  it  means  1  hundred,  showing  that  3  is  contained 
in  300,  100  times.  The  9  means  9  tens^  and  the  3 
which  we  place  under  it  means  3  tens,  showing,  that  3 
is  contained  in  90,  30  times. 

A  Dividend  is  a  number  which  is  to  be  divided;  such 
is  the  number  396  in  the  above  example.  A  Divisor  is 
a  number  by  which  we  divide;  such  is  the  number  3  in 
the  above  example.  The  Quotient  is  the  number  of 
times  which  the  divisor  is  contained  in  the  dividend; 
such  is  the  number  132  in  the  above  example. 

Find  the  quotient  m  each  of  the  following  examples. 

(2).  4)8        (3).  2)46        (4).  3)936        (5).  4)4884 


'    6.  A  man  laid  out  69  dollars  for  sheep,  paying  3  dol- 
lars a  head  for  them.     How  m.any  did  he  buy  ? 

7.   If  4  bushels  of  wheat  will  pay  for  1  barrel  of  flour, 
how  many  barrels  will  848  bushels  pay  for  ? 

Section   2. 
1.   How  many  times  is  4  contained  in  3684  ? 
4)3684  tn  this  example  we  find  that  4  is  not 

"  go  1       contained  in  3,  therefore  we  join  the  3 
' with  the  6,  and  say,  4  in  36,  9  times. 


.  2.  DIVISION.  109 

2.  How  many  times  is  7  contained  in  56? 

3.  How  many  times  is  9  contained  in  639? 

4.  How  many  times  is  5  contained  in  405? 

5.  How  many  times  is  4  contained  in  3248? 

6.  How  many  times  is  3  contained  in  1569?    * 

7.  If  4  horses  are  required  to  draw  1  wagon,  how 
many  wagons  might  be  drawn  by  168  horses? 

8.  How  many  yards  of  broad  cloth,  that  is  sold  at  6 
dollars  a  yard,  can  be  bought  for  492  dollars? 

9.  If  a  man  can  travel  5  miles   an   hour,  how  many 
hours  will  it  take  him  to  travel  205  miles? 

10.  Suppose  69  to  be  a  dividend,  and  3  a  divisor; 
what  is  the  quotient? 

11.  Suppose  128  to  be  a  dividend,  and  4  a  divisor; 
what  is  the  quotient? 

12.  Suppose  486  to  be  a  dividend,  and  6  a  divisor; 
what  is  the  quotient? 

13.  How  many  times  is  4  contained  in  872? 

4)8  7  2  4  in  8,  2  times;    4  in  7,   1  time,  and 

rrr       there  is  3  over;   (we  join  this  3  with  the  2, 
zll       making  32,)  then  4  in  32,  8  times. 

14.  How  many  times  is  6  contained  in  726? 

15.  How  many  times  is  8  contained  in  896? 

16.  How  many  times  is  5  contained  in  1605? 

17.  How  many  times  is  7  contained  in  924? 

18.  How  many  times  is  4  contained  in  6732? 

19.  Suppose  1585  to  be  a  dividend  and  5  the  divi- 
sor; what  is  the  quotient? 

20.  Suppose  4518  to  be  a  dividend  and  6  the  divi 
sor;  what  is  the  quotient  ? 

21.  How  many  times  is  7  contained  in  742? 

7)742  The  divisor  not  being  contained  once  in 
rr~  the  ten's  place  of  the  dividend  we  write  a 
0  in  the  ten's  place  of  the  quotient. 

22.  How  many  times  is  3  contained  in  609? 

23.  How  many  times  is  S  contained  in  1624? 

24.  How  many  times  is  5  contained  in  4015? 

25.  How  many  times  is  9  contained  in  2880? 

26.  How  many  times  is  7  contained  in  10500'* 

K 


110  WRITTEN   ARITHMETIC.  \\ 

27.  If  I  bad  78  dollars  to  lay  out  for  flour,  and  the 
flour  was  6  dollars  a  barrel,  how  many  barrels  could  1 
buy  for  all  the  money? 

23.  A  drover  received  268  dollars  for  sheep,  that  he 
sold  at  4  dollars  a  head.     How  many  were  there .'^ 

29.  If  1  ton  of  hay  be  worth  9  bushels  of  corn,  how 
many  tons  of  hay  are  576  bushels  of  corn  worth? 

30.  If  3  bushels  of  wheat  will  pay  for  a  yard  of  cloth, 
how  many  yards  will  105  bushels  pay  for? 

31.  How  many  soldiers  may  be  clothed  from  5708 
yards  of  cloth,  allowing  4  yards  to  make  a  suit? 

32.  How^  many  muskets  can  be  purchased  for  2952 
dollars;  the  price  being  6  dollars  apiece? 

33.  If  76  dollars  should  be  divided  equally  among  4 
men,  how  many  dollars  would  each  man  receive? 

If  there  were  only  4  dollars  to  be  divided,  each  man 
w^ould  receive  just  1  dollar:  therefore  each  man  must 
receive  as  many  dollars  as  there  are  fours  in  76. 

34.  Suppose  5  men  have  to  pay  a  bill  of  95  dollars, 
how  many  dollars  must  each  man  pay? 

35.  If  171  biscuit  be  divided  equally  among  a  crew 
of  9  sailors,  how  many  does  each  sailor  receive? 

36.  A  farmer  planted  354  trees,  in  6  equal  rows. 
How  many  were  there  in  1  row? 

37.  A  fisherman  hired  a  boat,  agreeing  to  give  the 
owner  1  fish  of  every  7  that  he  might  catch:  he  caught 
434.     How  many  should  he  give  the  owner? 

33.  8  sailors  received  1576  dollars  for  retaking  their 
ship.     How  much  did  each  sailor  receive? 

39.  A  man  intending  to  go  a  journey  of  336  miles, 
wishes  to  perform  it  in  6  days.  How  many  miles  must 
he  travel  each  day? 

40.  9  men  have  agreed  to  make  up  a  purse  of  2178 
dollars.     How  many  dollars  must  each  one  put  in? 

41.  Suppose  A  to  spend  3  dollars  as  often  as  B  spends 
1  dollar;  how  many  dollars  will  B  spend  while  A  is 
spending  89004  dollars? 

42.  Suppose  3656  dollars  have  been  equally  divided 
among  a  number  of  men,  and  each  man  has  received  8 
dollars;  how  many  men  were  there? 


2  DIVISION.  iil 

43.  A  number  of  men  contributed  9  dollars  apiece, 
and  thereby  made  up  a  purse  of  54  dollars.  How  many 
men  were  there  ? 

44.  Suppose  9  has  been  multiplied  by  some  number, 
and  the  product  is  54;  what  was  the  multiplier.'* 

45.  5  men  paid  equal  shares  of  a  debt  of  80  dollars. 
How  much  did  each  man  pay  .? 

46.  Suppose  some  number  has  been  multiplied  by  5, 
and  the  product  is  80;  what  number  was  multiplied  ? 

47.  Two  numbers  have  been  multiplied  together,  and 
their  product  is  126:  one  of  the  two  numbers  multiplied 
is  7; — what  is  the  other  ? 

48.  Divide  348  by  4;  then  prove  the  work  to  be  right, 
by  multiplying  the  quotient  and  divisor  together  } 

4)348  We  find  by  the  quotient,  there  are  87 

'~~Qj       times  4  in  348:  therefore  we  know  that  87 

^       times  4,  or  4  times  87,  must  make  348. 

■ Had  our  quotient  been  wrong,  our  product 

^^^       and  dividend  would  not  be  equal. 

49.  Divide  72  by  8,  and  prove  the  work  to  be  right 
60.   Divide  5890  by  5,  and  prove  the  work  to  be  right 

51.  Divide  39781  by  7,  and  prove  the  work  to  be  right. 

52.  Divide  90048  by  8,  and  prove  the  work  to  be  right. 

53.  Divide  17604  by  9,  and  prove  the  work  to  be  right. 

54.  A  hatter  has  130  hats  finished;  and,  in  order  to 
send  them  to  market,  he  must  pack  them  in  boxes,  that 
will  hold  8  hats  apiece.  How  many  full  boxes  can  he 
send;  and  how  many  hats  will  remain  on  hand  ? 

8)1  30  We  have  2  units  over.     This  2  is  a  re- 

1^2       'inainder;  it  shows  that  there  are  2  hats, 
• which  cannot  be  divided  into  eights. 

55.  How  many  sheep,  at  4  dollars  a  head,  can  a 
butcher,  who  has  747  dollars  buy;  and  how  many  dol- 
lars will  he  have  remaining  ? 

56.  If  5  yards  of  cloth  will  make  a  suit  of  clothes, 
how  many  suits  can  be  made  from  96  yards;  and  how 
many  yards  will  there  be  over  ? 

57.  How  many  times  is  6  contained  in  4637;  and 
how  many  are  there  over  ? 


112  WRITTEN   ARITHMETIC.  V. 

58.  How  many  times  is  8  contained  in    9150;   and 
how  many  are  there  over  ? 

59.  Suppose  568  to  be  a  dividend,  and  7  the  divisor; 
what  is  the  quotient,  and  the  remainder  ^ 

60.  Suppose  1953  to  be  a  dividend,  and  7  the  divi- 
sor; what  is  the  quotient,  and  the  remainder  ? 

61.  Divide  564  by  7,  and  prove  the  work  to  be  right. 
The  remainder,  in  division,  is  an  undivided  part  of  the 

dividend:  therefore,  the  remainder  must  be  added  to  the 
product  of  the  divisor  and  quotient,  to  make  the  product 
equal  to  the  dividend. 

62.  Divide  109  by  6,  and  prove  the  work  to  be  right. 

63.  Divide  817  by  5,  and  prove  the  work  to  be  right. 

Section  3. 
The  method  of  dividing  taught  m  the  two  preceding 
sections,  is  called  Short  division:  the  method  taught  in 
this  section;  is  called  Long  division.  In  long  division, 
we  place  the  quotient  on  the  right  hand  of  the  dividend, 
and  perform  some  operations  under  the  dividend,  here- 
tofore performed  in  the  mind. 

1.  How  many  times  is  4  contained  in  95307  ? 

tJ      b  ^  Perceiving  that  4  is  contained 

I]      I.*  §  in  9,  twice,  we  place  2  in  the 

§       §^  §"•         quotient,  multiply  the  divisor  by 

1,  S.         2,  and  subtract  the  product  (8) 

4)95307(23826     ^^^"^  ^-     This  is  the  same   as 

8  saying  in  short  division, '  4  in  9, 

"TT  2   times,  and    1   over.'     Now, 

y  2  since  the  1  over  must  be  joined 

. with  the  5,  we  bring  the  5  down 

33  to  the  right  of  the  1:  and  then, 

3^  perceiving  that  4  is  contained  in 

10  15,  3  times,  we  place  3  in  the 

8  quotient,    multiply   the    divisor 

*^  by  3,  and  subtract  the  product 

c^.  as  before.     Thus  we  proceed  to 

bring  down  every  figure  of  the 

Remainder  3  dividend,  and  unite  it  with  the 

previous  remainder. 


DIVISION  113 

Perform  the  following  examples  by  long  division. 

2.  How  many  times  5  are  there  in  7163  ? 

3.  How  many  times  7  are  there  in  88  704  ? 

4.  How  many  times  6  are  there  in  97  547  ? 

5.  How  many  times  3  are  there  in  8  057  251  ?     . 

6.  How  many  times  4  are  there  in  8  708  983  ? 

7.  How  many  times  5  are  there  in  6  457  080? 

8.  How  many  times  8  are  there  in  25  648  ? 

{c^c\c\r  The  divisor  noi  being  contain- 

8)25 648(3200       ^^  ^^^^  j^  ^^^^  l^^-^  j^^^j  ^^^^^ 

^^  of  the  dividend,  we  join  this  fig- 

1  6  ure  with  the  next.     After  bring- 

1  6  ing  down  the  4,  we  find  the  divi- 

'      ^  sor  is  not  contained  in  it;  there- 

^  g  fore,  we  place  a  0  in  the  quotient, 

■ —  and  bring  down  the  next  figure. 

9.  How  many  times  5  are  there  in  43  906  ? 

10.  How  many  times  9  are  there  in  70  223  ? 

11.  How  many  times  6  are  there  in  901  500  ? 

12.  How  many  times  7  are  there  in  161  635  ? 

13.  How  many  times  24  are  there  in  3762  ? 
24')3762fl56  "^^^^^  operation  is  performed 

'    .  in  the  same  manner  that  it  w^ould 

have  been,  if  the    divisor   had 

136  consisted  of  only  one  figure. 

1^0  The  two  following  examples 

152  will  show  the  method  of  deter- 

144  mining  when  a  figure  placed  in 

T^  the  quotient    is  too    great,   and 

—  when  it  is  too  small. 

1*4.  How  many  times  is  18  contained  in  12  532? 

In    this    example,    we    have 

1 8)12532(697        chosen  7  for  the  last  figure  of  the 

108  quotient;  but  it  appears,  that  7 

j^«-Q  times    18    are    more  than   112; 

^  />2  therefore  18  is  not  contained  7 

~ times  in  112.     Tht  7  and  the 

^  ^  ^  ^   product  arising  from  it  must  be 

^?.^  rubbed- out,  and  a  smaller  figure 

must  be  placed  in  the  quotient. 


114 

15. 


WRITTEN    ARITHMETIC. 


v: 

How  many  times  is  35  contained  in  45  817  ? 

Here  we  have  chosen  8  for 
the  last  figure  of  the  quotient; 


35)45817(1308 
35 


108 
105 

317 
280 

3? 

16.  How  many  times 

17.  How  many  times 

18.  How  many  times 

19.  How  many  times 
How  many  times 
How  many  times 
How^  many  times 
How  many  times 
How  many  times 


20. 
21. 

22. 
23. 
24. 


but,  after  subtracting  8  times 
35  from  317,  there  remains,  37. 
This  remainder  will  contain 
35,  once  more;  therefore,  we 
must  rub  out  the  8  and  the 
work  resulting  from  it,  and 
must  put  9  in  the  place  of  %S. 

s  47  contained  in  804  ? 
53  contained  in  1625  ? 

s  68  contained  in  94  605  } 

s  71  contained  in  661  419  ? 

s  108  contained  in  216  ? 

s  325  contained  in  7134.^ 

s  476  contained  in  92  107  ? 

s  504  contained  in  1008  ? 

s  651  contained  in  43  126  ^ 


RULE  FOR  DIVISION.  When  the  divisor  does  not  ex* 
ceed  9,  draio  a  line  tinder  the  dividend^  find  how  many 
times  the  divisor  is  contained  in  the  left  hand  figure^  or 
two  left  hand  figures  of  the  dividend^  and  write  the  figure 
expressing  the  number  of  times  underneath:  if  there  be  a 
remainder  over^  conceive  it  to  be  prefixed  to  the  next  fig-^ 
ure  of  the  dividend^  and  divide  the  next  figure  as  before* 

Thus  proceed  through  the  dividend. 

When  the  divisor  is  more  than  9,  find  how  many  times 
it  is  contained  in  the  fewest  figures  that  will  contain  it^ 
on  the  left  of  the  dividend^  write  the  figure  expressing 
the  number  of  times  to  the  right  of  the  dividend^  for  the 
first  quotient  figure;  multiply  the  divisor  by  this  figure^ 
and  subtract  the  product  from  the  figures  of  the  dividend 
considered.  Place  the  next  figure  of  the  dividend  on  the 
right  of  the  remainder^  and  divide  this  number  as  before. 

Thus  proceed  through  the  dividend. 
PROOF.     JVIultiply  the  divisor  and  quotient  together, 

and  to  the  product  add  the  remainder:  the  sum  will  bb 

mutl  to  the  dividsnd^  if  the  work  be  right* 


DIVISION  U5 
25.  Divide  46242  by  252,  and  prove  the  operation. 

252)46242(183  252 

252  183 

2104  756 

2016  2016 

882  ^^L^ 

756  ^^^ 


126 


46242 


26    Divide  74  201  by  625,  and  prove  the  operation. 

27.  Divide  408  732  by  9,  and  prove  the  operation. 

28.  Divide  15  362  by  88,  and  prove  the  operation. 

29.  Divide  57  026  by  492,  and  prove  the  operation. 

30.  Divide  982  700  by  53,  and  prove  the  operation. 

31.  Divide  162  941  by  256,  and  prove  the  operation* 

32.  Divide  648  035  by  14,  and  prove  the  operation* 

33.  Divide  106  401  by  333,  and  prove  the  operation* 

34.  Divide  62  509  by  4423,  and  prove  the  operation* 

35.  Divide  1  071  400  by  29,  and  prove  the  operation. 

36.  How  many  acres  of  land,  &t  22  dollars  an  acre^ 
can  be  bought  for  8514  dollars  ? 

37.  Suppose  a  man  to  earn  35  dollars  a  month;  how 
many  months  will  it  take  him  to  earn  490  dollars  ? 

38.  If  a  man  travel  48  miles  a  day,  in  how  many  days 
will  he  perform  a  journey  of  3264  miles  ? 

39.  If  774  dolhirs  be  divided  equally  among  IS  sail* 
ors,  how  many  dollars  will  each  sailor  receive  ? 

40.  If  a  man's  income  be  2555  dollars  a  year,  how 
much  is  it  a  day,  there  being  365  days  in  a  year  ? 

41.  The  income    of  the  Chancellor  of  England,  is 
99  280  dollars  a  year.     How  much  is  it  per  day  .'' 

42.  G3  gallons  of  water  will  fill  a  hogshead.     How 
4Tiany  hogsheads  will  5166  gallons  fill? 

43.  How  many  hogsheads  can  be  filled  from  19  721 
gallons  ? — and  how  many  gallons  will  there  be  left.'' 

44.  Suppose  a  regiment  of  512  men  have  8192  pounds 
of  beef;  how^  many  pounds  are  there  for  each  man } 

45.  If  a  dividend  be  46  319,  and  the  divisor  807^ 
what  is  the  quotient  ?  — and  what  the  remaiader  .'^ 


116  WRITTEN    ARITHMETIC.  V 

Section  4. 

ABBREVIATIONS. 

When  there  are  ciphers  on  the  right  hand  of  a  divisor,^ 
cut  them  off  J  and  omit  them  in  the  operation;  also  cut  off 
and  omit  the  same  number  of  figures  from  the  right  hand 
of  the  dividend.  Finally^  place  the  figures  cut  off  from 
the  dividend^  on  the  right  of  the  remainder. 

1.  How  many  times  900  are  there  in  741  725  ? 

.      9100)74 17125  ^^?  ^''^'^^  T^V  ^^  ^'  ^^'T 

• ^ remams  1 ,  to  which  we  annex  the 

824     125      25,  making  the  true  rem.  125. 

2.  How  many  times  70  are  there  in  8  563  512  ? 

3.  How  many  times  300  are  there  in  6374  ? 

4.  How  many  times  5000  are  there  in  46  578  ? 

5.  How  many  times  40  are  there  in  80  603  ? 

6.  How  many  times  600  are  there  675  700  ? 

7.  How  many  times  8000  are  there  in  16  000  ? 

8.  Divide  65  237  by  50,  and  prove  the  operation. 

9.  Divide  567  289  by  400,  and  prove  the  operation 

10.  How  many  times  570  are  there  in  35  871  ? 

11.  How  many  times  280  are  there  in  6423  ? 

12.  How  manv  times  4200  are  there  in  91  621? 

13.  How  many  times  9060  are  there  in  287  000  .? 

When  the  divisor  is  10,  100,  1000,  «^c.,  cut  off  as 
many  figures  from  the  right  hand  of  the  dividend^  as 
there  are  ciphers  in  the  divisor;  the  other  figures  of  the 
dividend  will  be  the  quotient^  and  the  figures  cut  off  will 
be  the  remainder. 

14.  How  many  times  10  are  there  in  240  ? 

15.  How  many  times  10  in  435;  and  how  many  over  } 

16.  How  many  times  100  are  there  in  4000? 

17.  How  many  times  100  in  748 ;  and  how  many  over  ? 

18.  100  cents  are  equal  to  1  dollar.  How  many 
dollars  are  there  in  5400  cents  ? 

19.  In  642  cents,  how  many  dollars  are  there;  and 
how  many  cents  over  ? 

20.  In  1937  cents,  how  many  dollars  are  there;  ami 
liow  many  cents  over  '^ 


4.  DIVISION.  117 

When  factors  of  the  divisor  can  be  founds  (that  i>, 
when  two  numbers  can  be  founds  which^  being  multiplied 
together^  produce  the  divisor^)  you  may  divide  the  divi- 
dend by  one  of  the  factors^  and  the  quotient  thence  arising 
by  the  other:  the  last  quotient  will  be  the  true  one, 

21.  In  a  certain  school  there  are  36  scholars,  among 
whom  540  quills  are  to  be  equally  divided.  How  many 
will  1  scholar  receive  ? 

Let  u&  suppose  the  school  to  be  divided  into  4  classes, 
allowing  9  scholars  to  be  in  each  class.  Then  we  will 
find  how  many  quills  1  class  will  receive,  and  from  this 
number,  find  how  many  1  scholar  will  receive. 

4)540     number  of  quills  for  the  school. 

9)135     number  of  quills  for  1  class. 

15     number  of  quills  for  1  scholar. 

Observe  in  the  above  example,  that  the  divisors  4  and 
9,  are  the  factors  of  36:  and,  if  we  had  divided  first  by 
the  9,  and  then  by  the  4,  our  last  quotient  would  have 
been  the  same  it  now  is. 

22.  Divide  11  376  by  72;  using  the  factors  of  72. 

23.  If  1024  dollars  be  divided  equally  among  64  men, 
how  many  dollars  will  1  man  receive  ? 

24.  How  many  times  is  42  contained  in  1176  i 

25.  If  27  yards  of  cloth  cost  216  dollars,  how  many 
dollars  does  1  yard  cost  ? 

26.  Suppose  1952  to  be  a  dividend,  and  32  the  divi- 
sor; what  is  the  quotient.'* 

To  obtain  the  true  remainder^  where  factors  have  been 
used  as  divisors^  multiply  the  last  remainder  by  the  first 
divisor^  and  to  the  product  add  the  first  remainder. 

27.  Suppose  622  to  be  a  dividend,  and  35  the  divi- 
sor; what  is  the  quotient;  and  what  the  remainder.'* 

28.  Suppose  99  to  be  a  dividend,  and  25  the  divisor; 
what  is  the  quotient;  and  what  the  remainder  ? 

29.  Suppose  4862  to  be  a  dividend,  and  81  the  divi- 
sor; what  is  the  quotient;  and  what  the  remainder  ? 

30.  Divide  1739  by  56. 


U8  WRITTEN    ARITHMETIC. 


Questions  to  be  answered  Orally, 
(!)  When  we  say,  ^3  is  contained  in  20,  6  times, 
and  2  over,'  which  of  these  numbers  have  we  for  the 
dividend  ? — Which  for  the  divisor  ? — Which  for  the 
quotient  ? —  Which  for  the  remainder  ?  (2)  What 
is  meant  by  the  dividend  ?  (3)  W^hat  is  meant  by 
the  divisor  ?  (4)  What  is  meant  by  the  quotient  ? 
(5)  What  is  meant  by  the  remainder  ?  (6)  Can  the 
remainder  ever  be  equal  to,  or  greater  than  the  divi- 
sor ? — Why  ?  (7)  Suppose  you  have  a  number  of 
dollars  to  divide  among  a  number  of  men;  which  num- 
ber do  you  make  the  dividend ; —  and  which  the  divisor  ? 
—  If  there  be  a  remainder,  will  it  be  so  many  dollars, 
or  so  many  men  ?  (8)  Recite  the  rule  for  division. 
(9)  How  do  you  proceed  when  there  are  ciphers  on 
the  right  hand  of  the  divisor?  (10)  How  do  you 
divide  by  10,  100,  1000,  &c.?  (11)  How  can  you 
divide  by  means  of  factors  ?  (12)  When  you  have 
divided  by  the  factors  of  the  divisor,  how  do  you  find 
the  true  remainder.'^  (13)  How  do  you  prove  an 
operation  in  division  ? 


Perform  the  following  examples  by  either  of  the  fore- 
going methods,  which  may  be  found  convenient. 

31.  Suppose  it  takes  7  bushels  of  apples  to  make  a 
barrel  of  cider,  how  many  barrels  of  cider  can  be  made 
from  945  bushels  of  apples  ? 

32.  Suppose  an  acre  of  land  properly  cultivated,  to 
produce  38  bushels  of  corn;  how  many  acres  must  be 
cultivated  to  produce  4902  bushels  ? 

33.  If  50  dollars  will  pay  for  an  acre  of  land,  how 
many  acres  can  be  bought  for  6900  dollars  ^ 

34.  How  many  days  will  a  ship  be  in  sailing  from 
New  York  to  Liverpool;  allowing  the  distance  to  be 
3000  miles,  and  the  ship  to  sail  100  miles  a  day  ^ 

35.  A  vintner  wishes  to  put  6615  gallons  of  wine  mto 
hogsheads  that  will  hold  63  gallons  apiece; — how  many 
hogsheads  must  he  have  ? 


4.  RETROSPECTIVE   OBSERVATIONS.  119 

36.  If  you  had  118  dollars,  how  many  hats  could  you 
pay  for,  at  5  dollars  apiece;  and  what  number  of  dollars 
would  you  have  left? 

37.  Suppose  a  drover  has  2130  dollars;  how  many 
oxen  can  he  pay  for,  at  47  dollars  apiece;  and  how 
many  dollars  will  he  have  left.^ 

38.  In  668  360  yards  of  cloth,  how  many  pieces,  and 
how  many  bales;  there  being  35  yards  in  each  piece, 
and  56  pieces  in  each  bale? 

39.  If  4810  dollars  be  shared  equally  among  130  men, 
how  much  will  each  man  receive? 

40.  A  farmer  planted  2072  trees  in  14  equal  rows. 
How  many  did  he  plant  in  a  row? 

41.  A  gentleman  wishes  to  spend  136  days  in  per- 
forming a  journey  of  3264  miles.  How  many  miles 
must  he  travel  each  day? 

42.  If  a  man  vvhose  property  is  valued  at  21 148  dol- 
lars be  worth  17  times  as  much  as  his  neighbor,  how 
much  is  his  neighbor  worth? 

RETROSPECTIVE    OBSERVATION'S. 

In  the  course  of  the  last  four  chapters,  you  have 
practised  four  kinds  of  operations  on  numbers:  viz. 
Addition,  Subtraction,  Multiplication,  and  Division. 
These  operations  should  be  perfectly  understood — the 
effect  of  each  should  be  distinctly  perceived;  for,  it  is 
on  their  proper  application,  that  the  solution  of  all  ques- 
tions in  arithmetic  depends. 

Addition  is  the  operation  by  which  two  or  more 
numbers  are  united  in  one  sum. 

Subtraction  is  the  operation  by  which  the  difference 
between  two  numbers  is  found. 

Multiplication  is  the  operation  by  which  a  number  is 
produced,  equal  to  as  many  times  one  given  number,  £s 
there  are  units  in  another  given  number. 

Division  is  the  operation  by  which  we  find  how  many 
times  one  number  contains  another, —  and,  by  which  we 
divide  one  given  number  into  as  many  equal  parts,  as 
there  are  units  in  another  given  number. 


120  WRITTEN  ARITHMETIC. 


Questions  to  be  answered  Orally* 
(1)  How  many  kinds  of  operations  are  practised 
on  numbers?  (2)  What  are  they  called?  (3)  What 
is  Addition?  (4)  What  is  Subtraction?  (5)  What 
is  Multiplication?  (6)  What  is  Division?  (7)  Pro- 
pose a  question  that  you  would  solve  by  addition. 
(8)  Propose  a  question  that  you  would  solve  by  sub- 
traction. (9)  Propose  a  question  that  you  would 
solve  by  multiplication.  (10)  Propose  a  questioti 
that  you  would  solve  by  division.  (11)  How  can  a 
question  in  multiplication  be  solved  by  addition? 
(12)  How  can  a  question  in  division  be  solved  by 
subtraction? 


Section  5. 

MISCELLANEOUS    EXAMPLES. 

1.  The  population  of  the  world  has  been  estimated 
to  be  as  follows.  North  America,  twenty-six  millions; 
South  America,  twelve  millions;  Europe,  two  hundred 
and  twenty  millions;  Asia,  five  hundred  millions;  Africa, 
thirty-eight  millions;  Australia,  four  millions.  What  is 
the  whole  number? 

2.  In  1830,  the  national  debt  of  the  United  States 
was  48  565406  dollars;  in  1831  it  was  39123191  dol- 
lars.    How  much  was  paid  in  one  year? 

3.  The  national  debt  of  England  cannot  be  less  than 
1900  000  000  dollars.  How  many  years  would  it  take 
to  pay  this  debt,  allowing  ten  millions  of  dollars  to  be 
paid  annually? 

4.  What  would  be  the  expense  of  laying  a  rail-way 
from  Louisiana  to  Maine;  the  distance  being  1800  miles, 
and  the  rail-way  costing  14  000  dollars  a  mile? 

5.  In  how  many  days  could  a  passage  be  effected 
from  Maine  to  Louisiana,  on  the  proposed  rail-way; 
allowing  a  car  to  run  25  miles  an  hour,  day  and  night? 

6.  How  many  days  would  it  take  a  man  to  ride  on 
horseback  from  Maine  to  Louisiana,  riding  5  miles  an 
hour,  and  10  hours  a  day? 


5  MISCELLANEOUS    EXAMPLES.  121 

7.  Light  passes  from  the  sun  to  the  earth — a  distance 
of  95  milhons  of  miles — in  about  8  minutes.  What 
distance  does  light  move  in  a  minute? 

8.  The  diameter  of  the  earth  is  7912  miles;  and  the 
diameter  of  the  sun  is  1 12  times  as  great.  What  is  the 
diameter  of  the  sun? 

9.  The  income  of  the  Bishop  of  Durham,  in  England, 
IS  106  560  dollars  per  annum.  How  many  clergymen 
would  this  support,  on  a  salary  of  800  dollars  per  annum? 

10.  Five  men  and  three  boys  found  a  sum  of  money, 
and  divided  it  so  that  each  man  had  43  dollars  and  each 
boy  26  dollars.     What  sum  did  they  find? 

11.  If  a  trader  buy  558  barrels  of  flour  at  5  dollars  a 
barrel,  and  pay  14  dollars  for  storage,  for  how  much 
must  he  sell  the  flour,  to  gain  160  dollars? 

12.  Suppose  5  bushels  of  wheat  to  make  a  barrel  of 
flour,  how  many  barrels  of  flour  can  be  made  from  12 
bins  of  wheat,  each  bin  containing  95  bushels? 

13.  In  12  times  95,  how  many  times  5? 

14.  If  a  farmer  sell  45  acres  of  land  at  38  dollars  an 
acre,  and  divide  the  money  equally  among  4  sons  and  1 
daughter,  what  is  each  one's  share? 

15.  A  man,  who  owned  520  acres  of  land,  purchased 
376  acres  more,  and  then  divided  the  whole  into  8  equal 
farms.     How  many  acres  did  each  farm  contain? 

16.  In  520  plus  376,  how  many  limes  8? 

17.  If  a  man's  income  be  1349  dollars  a  year,  and 
his  expenses  3  dollars  a  day,  how  much  will  he  lay  up 
in  a  year;  there  being  365  days  in  a  year? 

18.  A  merchant  gave  39  240  dollars  for  a  cargo  of 
sugar,  and  after  selling  it,  found  he  had  gained  1671 
dollars.     For  how  much  did  he  sell  it? 

19.  A  merchant  gave  18  dollars  a  hogshead  for  245 
hogsheads  of  molasses,  and  then  sold  the  whole  for  4000 
dollars:  did  he  gain  or  lose; — and  how  much? 

20.  A  lot  of  land  was  divided  into  8  farms,  and  each 
farm  contained  150  acres.  How  many  acres  were  there 
in  the  whole  lot? 

21.  If  a  man's  expenses  are  2  dpi.  a  day,  and  his 
income  17  dol.  a  week;  what  will  he  save  in  7  weeks? 

L 


122  WRITTEN    ARITHMETIC.  V. 

22.  Three  men  bought  a  ship :  the  first  man  paid  2274 
dollars;  the  second  paid  3  limes  as  much  as  the  first, 
and  the  third  paid  as  much  as  the  first  and  second  both. 
What  was  the  price  of  the  ship? 

23.  A  hogshead  holds  63  gallons.  How  many  gallons 
of  wine  are  there  in  20  hogsheads;  allowing  that  each 
hogshead  wants  5  gallons  of  being  full? 

24.  If  a  man  earn  36  dollars  a  month,  how  many 
months  will  it  take  him  to  earn  576  dollars? 

25.  If  a  man  earn  40  dollars  a  month,  and  spend  13 
dollars  a  month,  how  many  months  will  it  take  him  to 
lay  up  297  dollars? 

2G.  A  farmer  having  20  barrels  of  pork,  sold  9  barrels 
at  22  dollars  a  barrel,  and  the  remainder  at  19  dollars  a 
barrel.     What  did  he  get  for  the  whole? 

27.  If  a  trader,  who  has  152  barrels  of  flour,  should 
lay  out  1370  dollars  in  buying  more  flour,  at  5  dollars  a 
barrel,  how  many  barrels  would  he  have? 

28.  A  trader  hired  650  dollars,  and  in  6  months  paid 
all  but  92  dollars.      How  much  did  he  pay? 

29.  What  is  the  value  of  139  yards  of  broad-cloth,  at 
7  dollars  per  yard? 

By  the  method  of  reasoning  heretofore  practised,  we 
should  say  in  this  solution,  139  yards  are  worth  139 
times  7  dollars;  and  thus  w^e  should  make  7  the  multi- 
plicand, and  139  the  multiplier.  But  since  it  is  more 
convenient  to  make  the  smaller  number  the  multiplier,  we 
reason  thus, —  If  the  value  of  1  yard  were  1  dollar,  the 
value  of  139  yards  would  be  139  dollars;  since  the  value 
of  1  yard  is  7  dollars,  the  value  of  139  yards  is  7  times 
139  dollars:  and  accordingly  we  make  139  the  multipli- 
cand, and  7  the  multiplier. 

30.  A  trader  bought  240  sheep,  at  4  dollars  a  head, 
and  paid  for  them  in  cows,  at  20  dollars  a  head.  How 
many  cows  did  he  give? 

31.  If  I  pay  6  dollars  an  acre  for  the  ploughing  of  18 
acres  of  land,  and  100  dollars  for  having  the  whole 
planted  and  hoed,  what  does  the  cultivation  cost? 

32.  How  many  cows  at  19  dollars  a  head,  will  pay 
for  38  sheep  at  4  dollars  a  head? 


5.  MISCELLANEOUS   EXAMPLES.  123 

33.  A  farmer  bought  a  fields  valued  at  150  dols.,  for 
which  he  gave  9  cows,  valued  at  14  dols.  apiece,  and 
the  rest  in  money.     How  much  money  did  he  pay? 

34.  What  number  must  be  added  to  9  times  14,  m 
order  that  the  sum  shall  be  150? 

35.  If  a  stage  travel  13  miles  in  the  same  time  that  a 
wagon  travels  5  miles,  how  many  miles  will  a  stage 
travel  while  the  wagon  is  travelling  65  miles? 

36.  Suppose  that  9  bushels  of  wheat  will  fill  a  hogs- 
head; how  many  hogsheads  can  be  filled  from  a  heap 
containing  149  bushels;  and  how  many  bushels  will  be 
left  in  the  heap  ? 

37.  Charles  and  Joseph  are  studying  arithmetic. 
Charles  is  322  examples  in  advance  of  Joseph,  but 
Joseph  performs  55  examples  in  a  day,  and  Charles, 
41.     In  how  many  days  will  J.  overtake  C? 

3S.  Two  men  started  together  and  travelled  on  the 
same  road,  at  the  rate  of  7  miles  an  hour:  but  one  of 
them  rested  1  hour  in  every  3  hours,  and  the  other 
rested  1  hour  in  every  4  hours.  How  far  apart  were 
they,  at  the  end  of  12  hours? 

39.  A  drover,  having  599  dollars,  wishes  to  buy  all 
the  oxen  he  can  pay  for,  at  34  dollars  a  head,  and  then 
lay  out  the  remainder  of  his  money  for  sheep,  at  3  dollars 
a  headw     How  many  of  each  must  he  buy? 

40.  A,  B,  and  C  made  up  a  purse  of  500  dollars.  A 
put  in  16  dollars,  and  B  put  in  3  times  as  much.  How 
much  did  C  put  in? 

41.  A  merchant  bought  64  tons  of  hemp  at  215  dol- 
lars a  ton.  How  many  ten-dollar  bank  notes  did  it  take 
to  pay  for  the  hemp? 

42.  A  merchant  paid  9600  dollars  for  43  tons  of 
hemp  At  how  much  must  he  sell  the  hemp  per  ton,  in 
order  to  gain  247  dollars  ? 

43.  What  number  must  be  subtracted  from  7342,  in 
order  that  the  remainder  shall  be  456? 

44.  What  number  must  be  multiplied  by  30,  in  order 
that  the  product  shall  be  2130? 

45.  What  number  must  be  divided  by  15,  in  order 
that  the  quotient  shall  be  640? 


124  WRITTEN    ARITHMETIC.  V. 

Section    6. 

FEDERAL    MONEY. 

Federal  money  is  the  national  currency  of  the  United 
States.  Its  several  denominations  are, —  the  MILL,  the 
CENT,  the  DOLLAR,  and  the  EAGLE. 

10  mills  are  equal  in  value  to  1  cent. 

10  cents  are  equal  to  1  dime. 

10  dimes,  or  100  cents,  are  equal  to  1  dollar. 

10  dollars  are  equal  to  1  eagle. 
In  commerce,  we  express  eagles  in  dollars,  and  dimes 
in  cents.     For  example,  instead  of  saying,  2  eagles  and 
5  dollars,  we  say,  25  dollars:    and  instead  of  saying,  3 
dinpes  and  4  cents,  we  say,  34  cents. 

1.  How  many  cents  are  there  in  86  dollars?  (See 
method  of  multiplying  by  100,  in  page  106.) 

2.  How  many  cents  in  7  dollars  and  58  cents  .-^ 

3.  How  many  dollars  are  there  in  3700  cents  .^  (See 
method  of  dividing  by  100,  in  page  116.) 

4.  Howmany  do4s.  and  how  many  cts.  over,  in  534  cts.  ? 

This  character,  ^,  placed  before  a  number,  shows  the 
number  to  express  dollars.  For  example,  $12,  is  12 
dollars.  When  dollars  and  cents  are  expressed  in  one 
sum,  they  are  separated  by  a  point,  thus,  $4.16;  to  be 
read,  4  dollars  and  16  cents.  Observe,  there  must  be 
two  places  of  figures  for  cents:  therefore,  if  the  cents  be 
less  than  10,  a  cipher  must  be  placed  on  the  left  hand 
of  the  figure  which  expresses  them.  For  example,  56 
dollars  and  9  cents  is  written  thus,  $56.09. 

5.  What  is  the  whole  sum  of  $34.25,  $  18.04,  $  142, 
$176.81,  and  58  cents? 

34.25  ^^  writing  these  numbers  for  addition, 

1 8.04  ^^^  place  dollars  under  dollars,  and  cents 

142  under  cents.     We  then  add  up  each  col- 

1 7  6.8  1  umn,  just  as  we  add  the  columns  of  sim- 

^5  3  pie  numbers.     Finally,  we  point  off  two 

'             _  figures  on  the  right  of  the  sum  for  cents, 

^    71. oa  ^^^  ^Yie  other  figures  are  dollars. 


6  FEDERAL    MONEY.  196 

6.  What  IS  the  sum  of  $57.20,  $6.02,  and  $81.16.? 

7.  Add  together  $538,  $1.52,  $5.07,  and  63  cents. 

8.  Add  together  18  cents,  $70.19,  $56,  and  7  cents. 

9.  Add  together  36  dollars,  7  dollars  and  45  cents, 
86  cents,  130  dollars  and  6  cents,  and  340  dollars. 

JO.  Add  together  9  dollars,  1  dollar  and  70  cents,  13 
dollars  and  7  cents,  50  cents,  and  10  cents. 

11.  Add  together  47  cents,  62  dollars,  9  dollars  and 
12  cents,  5  dollars  and  5  cents,  and  3  dollars. 

12.  Add  together  37  dollars,  4  dollars  and  17  cents, 
96  dollars  and  1  cent,  99  cents,  and  2  dollars. 

13.  What  is  the  expense  of  one  quarter's  schooling, 
allowing  $  19  for  board,  $9  for  tuition,  $3.75  for  books, 
and  92  cents  for  stationary  ? 

14.  A  sailor  paid  $16.35  for  a  hogshead  of  molasses, 
in  New  Orleans,  and  also  paid  $3.40  for  the  freight  of 
the  molasses  to  Boston.  For  how  much  must  he  sell  it 
in  Boston,  in  order  to  gain  $4  .'^ 

15.  Subtract  $4.35  from  $6.48;  taking  cents  from 
cents,  and  dollars  from  dollars. 

16.  Subtract  $7.18  from  $48.50. 

17.  Subtract  $251.12  from  $546.18. 

18.  Subtract  $47.56  from  $319. 

319.00  ^^  writing  these  sums  of  money  for 

47.56  subtraction,  we  supply  the   places  of 

^'      1  A  A  cents  in  the  greater  sum,  by  ciphers, 

^ ! and  then  proceed  to  subtract. 

When  either  of  the  sums  of  Federal  money  presented 
for  subtraction  has  no  cents  expressed^  the  places  of  cents 
may  be  supplied  by  two  ciphers. 

19.  Subtract  $654  from  $783.48. 

20.  Subtract  $31.12  from  $5390. 

21.  Subtract  42  cents  from  $51. 

22.  Subtract  7  cents  from  $1. 

23.  Subtract  5  cents  from  $754. 

24.  Subtract  4  cents  from  $  1. 

25.  What  is  the  difference  between  $3.06,  And  $9  f 

26.  What  is  the  difference  between  $6,  and  7  cents? 


126  WRITTEN    ARITHMETIC.  V. 

27.  A  lady  having  $3,  paid  $1.15  for  a  yard  of  cam- 
bric.    How  much  money  had  she  left  ? 

28.  A  farmer  sold  a  barrel  of  pork  for  $21.50,  taking 
in  payment  a  hogshead  of  salt  at  $5,  and  the  rest  in 
money.     How  much  money  did  he  receive  ? 

29.  A  trader  began  business  with  $648,  and  at  the 
end  of  2  years,  had  $911.06.     What  did  he  gain? 

30.  A  traveller  having  no  money,  sold  his  horse  for 
$92.75,  and  his  gig  for  $78,  and  then  paid  $  17  for  pas- 
sage home.      How^  much  money  did  he  bring  home  ? 

31.  A  jockey  gave  $120  for  a  horse,  and  then  ex 
changed  for  another  horse,  receiving  $15.30  for  differ 
ence  of  value,  and  then  exchanged  again,  paying  $28.50 
How  much  did  the  last  horse  cost  him  ? 

32.  How  much  is  18  times  $4.72? 

$4.72  is  the  same  as  472  cents:  there  • 

4.72       fQYe  we  multiply  it  as  472  cents,  and  the 

^^       product  is  8496  cents.     Now  to  change 

37  7  6       these  cents  to  dollars,  we  must  divide  them 

47  2  by  100:  this  we  do,  by  pointing  off  two 

^^S4~96       figui'Gs  for  a  remainder.     The  quotient  is 

dollars,  and  the  remainder  is  cents. 

33.  How  much  is  4  times  $  1.08  ? 

34.  How  much  is  7  times  $52.31? 

35.  How  many  dollars  are  8  times  75  cents  ? 

36.  How  many  dollars  are  32  times  25  cents  ? 

37.  How  much  is  19  times  43  cents  ? 

38.  .How  much  is  241  times  $654.12? 

39.  What  is  the  value  of  6  pounds  of  Hyson  tea,  at 
$  1.20  cents  per  pound  ? 

40.  What  is  the  value  of  10  yards  of  flannel,  at  64 
cents  per  yard  ? 

41.  What  is  the  value  of  6  hats,  at  $6.47  apiece? 

42.  What  will  a  laborer  receive  for  25  days'  work, 
at  $1.15  per  day? 

43.  How  much  must  be  paid  for  30  pounds  of  coffee, 
when  the  price  is  16  cents  a  pound? 

44.  How  much  must  be  paid  for  12  drums  of  figs, 
when  the  price  is  $  1.55  a  drum  ? 


6  FEDERAL    MONEY.  1S7 

45.  What  is  the  value  of  147  bushels  of  apples,  at  8 
cents  per  bushel  ? 

It  is  more  convenient  in  this  example,  to  make  the 
number  of  bushels  the  multiplicand  and  the  number  of 
cents  the  multiplier.     For  method  of  reasoning,  see  re 
marks  in  section  5,  under  example  29.  ' 

46.  If  a  man  spend  2S  cents  a  day,  how  much  will 
he  spend  in  365  days,  or  1  year  ? 

47.  What  is  the  cost  of  430  pounds  of  chocolate,  at 
20  cents  per  pound  ? 

48.  At  6  cents  a  pound,  what  is  the  value  of  a  quarter 
of  beef,  weighing  214  pounds  ? 

49.  At  $2.30,  [230  cents]  an  acre,  what  is  the  value 
of  4748  acres  of  wild  land  ? 

When  the  price  of  a  single  article  is  given  in  Federal 
money^  and  the  value  of  any  number  of  that  article  is 
required^  either  the  price  may  be  multiplied  by  the  nttm- 
ber  of  articles  J  or  the  number  of  articles  by  the  price; 
the  product  loill  be  the  answer, 

50.  At  $  1.72  per  pound,  what  is  the  value  of  5  chests 
of  tea,  each  chest  containing  64  pounds.'^ 

51.  A  trader  gave  $5.16  a  barrel  for  2170  barrels  of 
flour,  and  sold  it  so  as  to  gain  $100.50  on  the  whole. 
For  how  much  did  he  sell  it  ? 

52.  A  mar>  bought  30  yards  of  cloth  at  $1.32  per 
yard,  and  30  yards  at  86  cents  per  yard.  How  much 
more  did  the  first  piece  cost,  than  the  last  ? 

53.  If  I  pay  22  cents  a  gallon  for  72  hogsheads  of 
rnolasses,  each  hogshead  containing  63  gallons,  and  then 
sell  the  whole  for  $936,  how  much  do  I  lose  } 

54.  A  man  having  $  350,  took  a  journey  of  700  miles, 
paying  6  cents  a  mile  for  stage  passage,  and  $  14  for 
board.     How  much  money  did  he  bring  home  ? 

55.  If  a  man  earn  $  1.02  a  day,  and  spend  36  cents  a 
day,  how  much  will  he  lay  up  in  75  days  .'' 

56.  If  a  man  get  §8.35  for  every  6  days'  work,  how 
much  will  he  get  by  working  510  days  ? 

57.  Suppose  42  casks  to  contain  46  gallons  of  wine 
.each;  what  is  the  value  of  the  whole,  at  $  1. 11  per  gal.  ? 


128  WRITTEN    ARITHMETIC.  V. 

58.  How  many  times  7  cents  are  there  in  $430.78? 

7)4  30.7  8  ^^  divide  5^430.78  as  if  the  figures 

stood  to  express  the  whole  in  cents. 

0  1  o  4       ryY^^  quotient  is  the  number  of  times. 

59.  How  many  times  6  cents  are  there  in  j^20.22? 

60.  How  many  times  15  cents  are  there  in  J^  11.10  ? 

61.  How  many  times  90  cents  are  there  in  ^27.00  ? 

62.  How  many  times  $4.06,  [406  cents,]  are  there  in 
$190  146.04,  [19014604  cents]  ? 

63.  How  many  lead  pencils  can  you  buy  for  $3.44, 
when  they  are  sold  at  8  cents  apiece  ? 

64.  How  many  pounds  of  butter,   at  21   cents    per 
pound,  can  be  bought  for  $3.57  ? 

65.  A  laborer  earned  $53.75,  by  working  at  $1.25  a 
day.     How  many  days  did  he  work  ? 

66.  If  84  cents  should  be  divided  equally  among  6 
boys,  what  would  each  boy  receive  ? 

67.  If  $28.71  [2871  cents]  be  divided  equally  among 
9  men,  what  will  each  man  receive  ? 

68.  If  $205  58  be  divided  equally  among  38  men, 
what  will  each  man  receive  ? 

69.  If  $637  be  divided  equally  among  24  men,  what 
will  each  man  receive  ^ 

After  dividing  the  number 
of  dollars  by  the  number  of 
men,  it  appears  from  the  quo- 
tient and  remainder,  that  each 
man  can  have  $26,  and  still 
$13  will  remain  undivided. 

We  change  $13  to  cents, 
by  annexing  two  ciphers,  and 
then  divide  the  cents  by  the 
number  of  men.  From  this 
quotient  and  remainder  it  ap- 
pears, that  each  man  will  have 
54  cents,  and  4  cents  will  re- 
main undivided. 

70.  If  $7640   be   divided   equally   among   61    men, 
what  will  each  man  receive  ? 


24)637(26 

48 

157 
144 

24)1300(54 
120 

100 
96 

4 

Answer y  $26.54. 
Remainder^  4  cents. 

6.  FEDERAL    MONEY  129 

71.  8  men  received  $230  for  performing  a  piece  of 
work.     What  was  each  one's  share  of  the  money  ? 

72.  An  insurance  office,  whose  stock  was  owned  in 
1000  shares,  divided  among  the  stock-holders,  §1586. 
How  much  was  paid  on  one  share  ? 

73.  The  expense  of  a  village  school,  for  6  months, 
was  5^466.80;  and  it  was  paid  in  equal  shares  by  40 
gentlemen.     What  was  each  one's  share  ? 

74.  Add  together  $9.87,  50  cents,  $705.30  and 
$390:  subtract  from  this  sum,  606  dollars  and  7  cents: 
multiply  the  remainder  by  45:  divide  the  product  by  37. 
What  is  the  quotient,  and  the  remainder  ? 

75.  'A  shoe-maker  paid  $1.58  apiece  for  10  calf-skins, 
and  22  cents  a  pound  for  3  sides  of  sole  leather,  each 
side  weighing  35  pounds.  From  this  stock  he  made  48 
pairs  of  shoes,  which  he  sold  at  $1.75  a  pair.  What 
did  he  get  for  his  work  ? 

76.  Suppose  a  man,  whose  income  is  $400  a  year, 
should  spend  $3.90  a  week,  how  much  would  he  save 
in  2  years;  there  boing  52  weeks  in  1  year  ? 

77.  Suppose  wheat  to  be  worth  $1.05  per  bushel, 
and  rye  70  cents  per  bushel:  how  many  bushels  of  rye 
must  be  given  for  550  bushels  of  wheat  ? 


Q^uestions  to  be  answered  Orally. 
(1)  What  is  Federal  money?  (2)  State  the 
denominations  of  Federal  money.  (3)  State  the 
number  of  mills  in  a  cent,  the  number  of  cents  in  a 
dime,  &c.  (4)  How  m_any  cents  make  a  dollar.'^ 
(5)  By  what  short  muthod  do  you  find  the  number  of 
cents  in  any  number  of  dollars  ?  (6)  How  do  you 
distinguish  the  number  of  dollars,  that  there  are  in 
any  number  of  cents?  (7)  in  writing  dollais  and 
cents  together,  how  many  figures  express  the  cents  ? 
(8)  When  the  cents  to  be  written  with  dollars  are 
less  than  10,  what  is  to  be  done  ?  (9)  Suppose  you 
are  dividing  dollars,  and  a  remainder  occurs,  what  is 
to  be  done,  in  order  to  divide  the  remainder  ? 


130  WRITTEN    ARITHMETIC.  V 

Section  7. 

TABLES  OF  COMPOUND  NUMBERS. 

ENGLISH  MONEY  is  the  national  currency  of  England 
4    farthings  (qr.)  ....  make  1  penny.  d 

12    pence make  1  shilHng.  s. 

20    shillings make  1  pound.  £. 

TROY  WEIGHT  is  used  in  weighing  gold  and  silver. 

24    grains  (gr.) make  1  pennyweight.        dwt 

20    pennyweights   ....  make  1  ounce.  oz 

12    ounces make  1  pound.  lb. 

AVOIRDUPOIS  WEIGHT  is  the  common  weight,  used  in 
weighing  groceries,  and  all  coarse  commodities. 

16    drams  (dr.) make  1  ounce.  oz* 

16    ounces make  1  pound.  lb. 

28    pounds make  1  quarter.  qr. 

4    quarters make  1  hundred-weight.  cwt« 

20    hundred-weight  .  .  .  make  1  ton.  T. 

APOTHECARIES'  WEIGHT  is  used  for  the  purpose  of 
compounding  medicines,  but  not  in  selHng  them. 
20    grains  (gr.) make  1  scruple.  9 

3  scruples make  1  dram.  5 

8    drams make  1  ounce.  § 

12    ounces make  1  pound.  ib 

CLOTH  MEASURE  is  used  in  measuring  cloth,  ]ace,  &c. 

4  nails  (na.) make  1  quarter.  qr. 

4  quarters make  1  yard.  yd- 

5  quarters make  1  Enghsh  ell.  E.  e 

5    quarters make  1  French  ell.  Fr.  e. 

3  quarters make  1  Flemish  ell.        Fl.  e. 

DRY  MEASURE  is  used  in  measuring  grain,  salt,  &c. 

2    pints  (pt.)    make  1  quart.  qt- 

8    quarts    make  1  peck.  pk 

4  pecks         make  1  bushel.  bu 


7  COMPOUND    NUMBERS.  131 

WINE  MEASURE  is  used  by  grocers  and  others,  for 
measuring  wine,  oil,  molasses,  and  most  other  liquids. 

4    gills  (gi.) make  1  pint.  pt 

2    pints make  1  quart.  qt. 

4    quarts make  1  gallon.  gal. 

31^-  gallons make  1  barrel.  bi. 

42    gallons make  1  tierce.  tier. 

63    gallons make  1  hogshead.  hhd, 

84    gallons make  1  puncheon.  pun. 

126    gallons make  1  pipe  or  but.  p. 

2    pipes,  or  4  hlids.  .  .  make  1  ton.  T. 

BEER  MEASURE  is  used  in  measuring  malt  liquors 

2    pints  (pt.) make  1  quart.  qt. 

4    quarts make  1  gallon.  gal. 

9    gallons make  1  firkin.  fir. 

2    firkins make  1  kilderkin.  kil 

2  kilderkins make  1  barrel.  bl. 

LONG  MEASURE  is  applied  to  length,  distance,  &c. 

3  barley-corns make  1  inch.  in. 

12    inches make  1  foot.  ft. 

3    feet make  1  yard.  yd. 

5^  yards  or  16^  feet    .  .  make  1  rod  or  pole.  r. 

40    rods make  1  furlong.  fur. 

8  furlongs    make  1  mile.  m. 

3  miles    make  1  league.  1 

9f  furlongs make  1  geographical  mile. 

60    geographical  miles  .  make  1  degree.  deg. 

360    degrees the  earth's  circumference. 

SQUARE  MEASURE  is  used  in  measuring  land,  floor- 
ing, boards,  tihng,  and  all  other  surfaces  whatever. 
144    inches make  1  foot.  ft. 

9  feet make  1  yard.  yd. 

30^-  yards,  or  272^-  ft.  .  .  make  1  rod  or  pole.  r. 

40    rods make  1  rood.  R. 

4  roods make  1  acre.  A. 

640    acres ^ake  1  mile  ml. 


132 


WRITTEN    ARITHMETIC. 


CUBIC  MEASURE  is  used  in  measuring  solid  bodies, 
and  in  finding  the  capacity  of  rooms,  boxes,  &c. 

1728    inches make  1  foot.  ft. 

40    feet  of  round  timber  .  make  J  ton.  T. 

50    feet  of  hewn  timber  .  make  1  ton.  T. 

16    cubic  feet make  1  foot  of  wood.  ft.w. 

8    feet  of  wood make  1  cord  of  wood.  C 

TIME  is  naturally  divided  into  days^  by  the  revolution 
of  the  earth  upon  its  axis;  and  into  years j  by  the  revo- 
lution of  the  earth  round  the  sun. 

60    seconds make  1  minute.  m. 

60    minutes make  1  hour.  h. 

24    hours make  1  day.  d 

365    days make  1  year.  Y. 

The  earth  revolves  round  the  sun  once  in  365  days, 
5  hours',  48  minutes,  and  48  seconds:  this  period  is 
therefore  a  Solar  year.  In  order  to  keep  pace  with  the 
solar  year  in  our  reckoning,  we  make  every  fourth  year 
to  contain  366  days,  and  call  it  Leap  year. 

The  year  is  divided  into  12  months.  The  number 
of  days  in  each  month  is  commonly  learned  thus, — 


'30  days  hath  September, 
April,  June,  and  November ; 
February  hath  28  alone, 
And  all  the  rest  have  31. 
Leap  year  comes  1  year  in  4 ; 
Then  February  hathl  day  more.' 


'The  4th.  11th.  9th.  and  6th., 
Have  30  days  to  each  affixed ; 
And  every  other,  31, 
Except  the  2nd.  month  alone, 
Which  has  but  28,  in  fine, 
Till  leap  year  gives  it  29.' 


Questions  to  he  answered   Orally. 
(1)   What  is  English  money  .^       (2)   Recite  the  ta- 
ble,      (3)   How  many  shillings  are  there  in  2  pounds  ^ 
(4)    How  many  pence  in  2  shillings  ?     (5)   How  ma- 
ny pence  in  8  farthings  }       (6)   How  many  pence  in 
36  farthings  }       (7)   How  many  farthings  in  6d.  3qr.  ^ 
(8)  Wha^  is  the  use  of  Troy  Weight.^       (9)   Re- 
cite the  table.       (10)   How  many  ounces  in  2  pounds  .^ 
j  (11)  How  many  penny-weights  in  4  ounces  ^ 


f  COMPOUND     NUMBERS.  133 

(12)  What  is  the  use  of  Avoirdupois  Weight? 
(13)  Recite  the  table.  (14)  How  many  ounces  in  4 
pounds  ?  (15)  How  many  hundred-weight  in  3  tons  } 
(16)   How  many  hundred-weight  in  24  quarters.^ 

(17)  How  is  Apothecaries' Weight  used  ?  (18) 
Recite  the  table.  ^19)  How  many  scruples  m  4 
drams  ?       (20)   How  many  drams  in  27  scruples  ? 

(21)  What  is  the  use  of  Dry  Measure  ?  (22)  Re- 
cite the  table.  (23)  How  many  pecks  in  12  bushels 
and  2  pecks  ?  (24)  How  many  bushels,  and  how 
many  pecks  over  in  35  pecks  ? 

{2o)  What  is  the  use  of  Cloth  Measure  .?  (26) 
Recite  the  table.  (27)  In  5  yards  how  many  quar- 
ters ?  (2S)  In  32  nails  how  many  yards  ?  (29) 
How  many  English  ells  in  14  quarters  ? 

(30)  To  what  is  Wine  Measure  ap'plled  ?  (31) 
Recite  the  table.  (32)  In  2  gallons  of  vinegar  how 
many  quarts  ? — how  many  pints  ? — how  many  gills  ? 
(33)    In  32  gills  how  many  gallons  ? 

(34)  What  is  measured  by  Beer  Measure?  (35) 
Recite  the  table.  (36)  How  many  gallons  are  there 
in  6  firkins  ?  (37)  How  many  kilderkins  in  19  fir- 
kins ?       (33)   How  many  frrkins  in  1  barrel  ? 

(39)  What  is  Long  Measure  applied  to  ?  (40) 
Recite  the  table.  (41)  What  number  of  inches  are 
there  in  1  yard  ?     (42)   How  many  feet  in  38  inches  ? 


(43)   In  6  furlongs  how  many  miles  ? 

(44)  What  is  the  use  of  Square  Measure  ?  (45) 
Recite  the  table.  (46)  How  many  square  feet  in  4 
square  yards  ?  (47)  How  many  rods  in  2  roods  ? 
(48)   How  many  acres  in  20  roods  ? 

(49)  What  is  the  use  of  Cubic  Measure  ?  (50) 
Recite  the  table.  (51)  How  many  cubic  feet  are 
there  in  3  feet  of  wood  ?  (52)  In  48  feet  of  wood, 
how  many  cords  of  wood  ? 

(53)  How  is  time  naturally  divided?  (54)  Re- 
cite the  table.  (55)  Wfiat  is  a  Solar  year  ?  (56) 
How  many  'months  in  a  year  ?  (57)  Recite  the 
lines  that  tell  the  number  of  days  in  each  month. 


134 


WRITTEN    ARITHMETIC 

Section   8. 


V. 


REDUCTION  OF  COMPOUND  NUMBERS. 
Reduction  is  the  operation  of  changing  any  quantity 
from  its  number  in  one  denomination,  to  its  number 
in  another  denomination.  For  ^"nstance,  if  we  change 
an  admeasurement  from  2  feet  to  24  inches,  that  is,  if 
we  find  how  many  inches  there  are  in  2  feet,  the  opera- 
tion is  called  reduction.  Again,  if  we  change  24  inches 
to  2  feet,  this  operation  is  also  called  reduction. 

ENGLISH    MONEY. 

2.  How  many  pounds  in 
128S2  farthings  ? 


many   farthings 


1.  How 
in  Jei3  8s.  4d.  2qr. 

£    s.  d 
13    8    4 
20 


qr. 

2 


268    shillings. 
12 


3220 
4 


pence. 


Ans.    12882    farthings.  Jins. 

In  this  example,  we  con- 
sider, that  there  are  20 
times  as  many  shillings  as 
pounds  in  any  sum;  there- 
fore we  multiply  the  13 
pounds  by  20,  and  add  the 
S  shillings  to  the  product. 
Then,  since  there  are  12 
times  as  many  pence  as 
there  are  shillings,  we  mul- 
tiply the  shillings  by  12  and 
add  the  4  pence  to  the  pro- 
duct. Lastly,  since  there  ,the 
are  4  times  as  many  farth- 
mgs  as  pence,  we  multiply 
the  pence  by  4,  and  add  2 
farthings  to  the  product. 


qr. 

4)12882 

1 2)3220    2qr. 
2|0)26i8  4d. 
8s. 


£13 


£ 
13 


d.  qr. 

4   2 


This  example  is  the  re- 
verse of  the  first  example. 
We  here  consider,  that  ev- 
ery 4  farthingS:  make  1  pen- 
ny; therefore,  we  find  by 
division  how  many  times  4 
there  are  in  the  number  of 
farthings:  the  quotient  is 
pence,  and  the  remainder  is 
farthings.  Then,  since  ev- 
ery 12  pence  is  1  shilling, 
we  divide  the  pence  by  12; 
quotient  is  shillings, 
and  the  remainder  pence. 
Lastly,  since  every  20  shil- 
lings make  1  pound,  we  di- 
vide the  shillings  by  20 


8 


COMPOUND    NUMBERS. 


135 


RULE  FOR  REDUCTION.  When  a  greater  denomina- 
tion is  to  be  reduced  to  a  smaller^  multiply  the  greater 
denomination^  by  that  number  which  is  required  of  the 
smaller^  to  make  ONE  of  the  greater;  adding  to  the 
product  so  many  of  the  smaller  denomination  as  are  ex- 
pressed in  the  given  sum.  Perform  a  like  operation  on 
this  product^  and  on  each  succeeding  product. 

When  a  smaller  denomination  is  to  be  reduced  to  a 
greater^  divide  the  smaller  denomination  by  that  number 
which  is  required  of  the  smaller^  to  make  ONE  of  the  next 
greater:  the  quotient  will  be  of  the  greater  denomination^ 
and  the  remainder  will  be  of  the  same  denomination  with 
the  dividend.  Perform  a  like  operation  on  this  quotient, 
and  on  each  succeeding  quotient, 

3.  How  many  farthings  are  there  in  18s.  7d.  3qr.  ? 

4.  How  many  pounds  are  there  in  9207  farthings  ? 

5.  How  many  pence  are  there  in  £5   Os.  lid.  ? 

6.  How  many  shilhngs  are  there  in  647  farthings .'' 

7.  How  many  times  8  pence  are  there  in  £3  5s.  } 

TROY    WEIGHT. 


8. 
1 51b 


How  many  grains  in 
lloz.  18dwt? 


lb. 
15 
12 

191 
20 

3838 
24 

15352 
7676 


oz. 
11 


dwt. 
18 


•5ns.   92112   grains 


9.  How  many  pounds  in 
92112  grains? 

2|0) 
24)92112(38318 


72 

201 

192 


91 

72 

192 
192 


191     18 

12)191(15 
12 

71 
60 

11 


Ans,  151b.  lloz.  18dwt 

10.  How  many  penny-weights  in  91b.  13oz.  16dwt.  / 

1 1 .  How  many  pounds  of  silver  in  829  penny-weights  ^ 

12.  How  many  grains  in  lOoz.  19dwt.  12gr.  ? 

13.  How  many  pounds  in  23641  grains  ? 


136  WRITTEN    ARITHMETIC.  V. 

AVOIRDUPOIS    WEIGHT. 

14.  How  many  pounds  are  there  in  1  ton  ? 

15.  How  many  drams  are  there  in  7  tons,  3  quarters, 
27  pounds,  5  ounces,  and  18  drams  ? 

16.  How  many  tons  are  there  in  31122  pounds  ? 

17.  What  will  3  hundred-weight,  3  quarters,  and  17 
pounds  of  indigo  cost,  at  ^2.67  per  pound  ? 

18.  A  wealthy  farmer  wishes  to  put  down  3T.  IScwt. 
2qr.  81b.  of  butter,  in  firkins,  containing  50  pounds 
apiece.     How  many  firkins  will  it  require  ? 

apothecaries'  weight. 

19.  How  many  scruples  are  there  in  1  pound .'' 

20.  How  many  pounds. are  there  in  1395  drams  ? 

21.  In  3ib  9§  05  19  lOgr.  of  epecacuanha,  how 
many  doses  are  there;  each  dose  containing  30gr.  .'* 

22.  If  it  take  1  ounce  of  salts  for  a  dose,  what  will 
75  pounds  amount  to,  at  4  cents  a  dose  ? 

23.  If  it  take  10  grains  of  calomel  and  1  scruple  of 
jalap  for  a  dose,  how  many  doses  are  there  in  iBb  15 
45  of  such  a  mixture  ? 

cloth  measure. 

24.  How  many  nails  are  there  in  1  English  ell  ? 

25.  How  many  yards  are  there  in  16240  nails  ? 

26.  In  320  yards,  and  3  quarters,  how  many  quarters  ? 
How  many  Flemish  ells  ? 

27.  How  many  more  nails  are  there  in  75  English 
ells,  than  there  are  in  93  yards  ? 

28.  A  shop-keeper  sold  cloth  enough  in  one  day  to 
gain  £6  Is.  8d.,  at  a  profit  of  2  farthings  on  every 
yard.     How  much  did  he  sell  ?  , 

DRY    MEASURE. 

29.  How  many  pints  are  there  in  1  bushel  ? 

30.  How  many  pints  are  there  in  58  bushels,  3  pecks, 
7  quarts,  and  1  pint? 

31 .  How  many  bushels  are  there  in  8240  quarts  ? 

32.  If  3  bushels  and  2  pecks  of  corn  will  fill  a  barrel, 
what  quantity  of  corn  will  20  barrels  hold  ? 

33.  Suppose  it  takes  3  pecks  of  salt  to  preserve  a 
barrel  of  pork,  how  much  salt  would  be  necessary  to 
preserve  351  barrels  of  pork? 


8.  COMPOUND    NUMBERS.  137 

WINE    MEASURE. 

34.  How  many  gills  are  there  in  1  hogshead  ? 

35.  How  many  hogsheads  are  there  in  90S4  pints  ^ 

36.  If  3  tierces  of  molasses  be  sold  at  12  cents  a 
quart,  what  will  the  whole  amount  to  ? 

37.  What  would  2  pipes  of  Madeira  wine  amount  to, 
at  67  cents  per  quart  ? 

3S.  A  certain  toper  drank  1  gill  of  rum  every  forenoon, 
and  1  in  the  afternoon,  for  6  years;  in  consequence  of 
which,  he  died.     How  many  hogsheads  did  he  drink  ? 

BEER    MEASURE. 

39.  In  2  barrels  and  1  firkin,  how  many  pints  } 

40.  In  6538  quarts,  how  many  kilderkins  ? 

41.  How  many  bottles,  holding  6  gills  apiece,  will  be 
required,  to  bottle  6  barrels  of  porter  ? 

42.  A  man  retailed  4  barrels  of  ale,  and  received  for 
it  $69.12.     At  what  price  did  he  sell  it  a  pint  .'^ 

43.  Suppose  a  retailer  to  sell  3  quarts  of  porter  every 
day  for  1  year,  excepting  52  Sabbaths,  how  many  bar- 
rels would  he  sell  in  the  year } 

LONG  MEASURE. 

44.  In  35  yards,  2  feet,  10  inches,  how  many  inches  .'* 

45.  In  29578  barley-corns,  how  many  yards  ? 

46.  In  16  leagues  and  2  miles,  how  many  rods  ? 

47.  Ho\V  many  geographical  miles  would  a  ship  sail, 
in  going  round  the  globe  ? 

48.  In  2541  inches  of  wire,  how  many  yards? 

49.  Suppose  7  inches  of  wire  to  make  1  link  of  a 
chain,  and  4  links  to  measure  1  foot;  how  many  yards 
of  wire  would  make  a  chain  8  feet  long } 

SQUARE    MEASURE. 

To  find  the  number  of  square  inches^  feet^  or  rods^  in 
any  surface  wJiich  has  four  sides^  and  four  equal  angles^ 
[corners^']  multiply  the  length  and  breadth  together, 

50.  How  many  square  inches  are  there  in  a  slate, 
that  is  13  inches  long,  and  8  inches  wide  ? 

51.  How  many  square  rods  are  there  in  a  field  28 
rods  long,  and  16  rods  wide  ?     How  many  acres  ^ 

52.  How  many  square  yards  of  carpeting  will  cover  a 
floor  36  feet  long,  and  18  feet  wide } 


138  WRITTEN    ARITHMETIC.  V 

CUBIC    MEASURE. 

A  cube  may  be  illustrated  by  a  solid  block,  having  6 
equal  sides.  Let  us  suppose  we  have  before  us  a  num- 
ber of  small  blocks,  representing  cubic  inches.  If  we 
lay  144  of  these  blocks  together  upon  the  table,  they 
will  cover  a  square  foot.  Then,  if  we  cover  this  layei 
of  blocks  with  another  layer,  and  thus  continue  till  wf« 
have  piled  up  12  layers,  the  pile  will  contain  12  times 
144  cubic  inches,  or  1  cubic  foot.  Therefore,  to  find 
the  cubical  contents  of  any  things  multiply  its  lengthy  and 
breadth  J  and  depth  together, 

53.  How  many  cubic  inches  are  there  in  a  brlck^ 
that  is  8  inches  long,  4  inches  wide,  and  2  inches  thick  ? 

54.  How  many  cubic  feet  in  a  box,  that  is  25  inches 
long,  20  inches  broad,  and  11  inches  deep  .^ 

55.  How  many  cubic  inches  in  1  ton  of  hewn  timber  ? 
66,   How  many  cubic  feet  in  a  pile  of  wood  15  feet 

long,  4  feet  wide,  and  5  feet  high .''     How  many  feet  of 
wood  .'*     How  many  cords  ? 

57.  How  many  cubic  feet  in  a  cord  of  wood  ? 

TIME. 

58.  How  many  seconds  are  there  in  a  common  year  .^ 
How  many  in  a  leap  year  ?     How  many  in  a  solar  year  ? 

59.  How  many  minutes  are  there  in  57  days  ? 

60.  If  your  pulse  beat  73  times  in  a  minute,  how 
many  times  will  they  beat  in  the  month  of  January  ? 

61.  How  many  years  and  days,  from  the  1st  day  of 
January,  1830,  to  the  1st  day  of  October,  1834  ? 


Questions  to  be  answered  Orally. 
(1)  What  is  meant  by  reduction  ?  (2)  How  do 
you  reduce  shilHngs  to  pence  ?  (3)  How  do  you 
reduce  pence  to  shillings  r  (4)  How  do  you  reduce 
Avoirdupois  ounces  to  pounds  ? — Why?  (5)  How 
do  you  reduce  pounds  to  ounces  ? —  Why  ?  (6) 
How  do  you  reduce  yards  to  nails  ?  (7)  How  do 
you  reduce  nails  to  yards  ?  (8)  Recite  the  general 
rule  for  reduction. 


1    w 

hat  is  the  wl 

I2s.  Od. 

Iqr.,   £60 

£7  10s. 

,  lOd.,  Is.  8( 

£. 

s.     d.     qr 

13 

7    10    2 

4 

12      0     1 

60 

0    11     3 

19      0     2 

116 

0      0     0 

7 

10    10     0 

1       8     3 

76 

0      0     0 

278 

12      5     3 

COMPOUND    NUMBERS.  139 

Section   9. 

COMPOUND    ADDITION. 
ENGLISH    MONEY. 

lole  sum  of  £13  7s.  lOd.  2qr.,  £4 
Os.  lid.  3qr.,  19s.  Od.  2qr.,  £116, 
i.  3qr.,  and  £76?    . 

The  sum  cf  the  column  of 
farthings  is  11 ;  equal  to  2d.  3qr. 
We  write  the  3qr.  and  add  the 
2d.  to  the  column  of  pence. 
The  sum  of  the  pence  is  41; 
equal  to  3s.  5d.  We  write  the 
5d .  and  add  the  3s.  to  the  column 
of  shillings.  The  sum  of  the 
shillings  is  52;  equal  to  £2  12s. 
We  write  the  12s.  and  add  the 
£2  to  the  column  of  pounds. 

RULE  FOR  COMPOUND  ADDITION.  Write  the  num- 
bers so  that  each  denomination  shall  stand  in  a  separate 
column.  Md  the  numbers  of  the  lowest  denomination 
together^  and  divide  their  sum  by  that  number  which  is 
required  of  this  denoviination  to  make  1  of  the  next  high- 
er: write  the  remainder  under  the  column  added^  and 
.carry  th^  quotient  to  the  next  column.  Thus  proceed 
with  every  denomination. 

2.  What  is  the  sum  of  £4  18s.  9d.,  £100  7s.  Od. 
Iqr.,  16s.  4d.,  3s.  6d.  2qr.,  £20,  and  £9   7s.  4d. .? 

3.  What  is  the  sum  of  Us.  Od.  3qr.,  £33  2s.  6d., 
8s.  7d.  Iqr.,  £450,  £9  17s.  8d.  3qr.,  and  £37  9s..?' 

4.  A  man  in  London  paid  for  a  hat,  £1  18s.  6d.;  for 
a  coat,  £9  8s.  4d.,  for  a  vest,  £1  10s.;  for  pantaloons, 
£3;  for  boots,  £1  2s.     What  did  the  suit  cost.^ 

TROY    WEIGHT. 

5.  Add  together  these  quantities  of  silver.  4lb.  9oz. 
16dwt.,  lOoz.  Idwt.  22gr.,  and  31b.  4oz.  Odwt.  6gr. 

6.  Add  together  lloz.  15dwt.  18gr.,  21b.  lOoz.  ISdwt. 
23gr.,  91b.  Ooz.  17dwt.  3gr     and  5oz-  12dwt 


140  WRITTEN    ARITHMETIC.  V. 

AVOIRDUPOIS    WEIGHT. 

7.  Add  together  14T.  lOcwt.  2qr,  231b.  4oz.,  27T. 
4cwt.  2qr.  241b.  14oz.,  and  3qr.  01b.  15oz.   lldr. 

C.  Add  together  16cwt.  Iqr.  111b.   6oz.  IGcvvt.  2qr. 
201b.,  5T.  Ocvvt.  3qr.  5lb.'  13oz.  2dr.,  and  2T. 
apothecaries'  weight. 

9-  What  is  the  weight  of  a  mixture  containing  5Bb 
log  53  19  8gr.,  63  29,  53  19  18gr.,  and  2ft>  4§  .^ 

10.  What  is  the  weight  of  a  mixture  containing  1  ii 
3§  13  29,  75  53  19  15gr.,  and  4  Jb  0§  63  ? 

cloth  measure. 

11.  Add  together  19yd.  2qr.   3na.,  14yd.  2qr.  Ina., 
32yd.  Oqr.  Ina.,  2qr.  2na.,  and  57yd.  3qr.  2na.' 

12.  Add  together  15E.e.  4qr.  2na.,  6E.e.  3qr.  Ina., 
45E.e.  3qr.  3na.,  230E.e.,  and  4E.e.  4qr. 

d'ry  measure. 

13.  Add  together  25bu.  2pk.  5qt.,  240bu.  Opk.  6qt , 
316bu.  3pk.  7qt.  Ipt.,  and  650bu.  2pk.  5qt. 

14.  Add  together  635bu.  Opk.  3qt.,  247bu.  3pk.  Oqt. 
Ipt.,  2bu.  3pk.  6qt.,  56bu.,  and  31bu.  Opk.  2qt. 

wine  measure. 

15.  How  many   hogsheads    are   12hhd.    42gal.    3qt. 
Ipt.,  548hhd.  62gal.  3qt.,  and  Shhd.  9gal.  Iqt.  ? 

16.  How  many  tons   are  IT.   Ip.   116gal.   3qt.,   Ip 
48gal.,  5T.  Ip.  86gal.  3qt.,  102gal.,  and  4T.  ? 

BEER    MEASURE. 

17.  Add  together  5bl.  Ikil.  Ifir.  8gal.  3qt.,  Ifir.  5gal. 
2qt.  Ipt.,  16bl.  Okil.  Ofir.  4gal.,  and  25bl.  Ikil. 

18.  Add  together  Ifir.  7gal.    3qt.,    24bl.    Okil.   Ifir. 
6gal.  2qt  Ipt.,  and  20bl.  Ikil.  Ifir.  4gal. 

LONG    MEASURE. 

19.  How  many  yards  are  45yd.  2ft.  llin.,  13yd.  Oft. 
9in.,  1ft.  lOin.,  and  20yd.  ift.  Sin.  2b.c.  ? 

20.  How  many  miles   are    10m.   Ofur.   36rd.,   58m. 
7fur.  13rd.,  38rd.,  16m.  4fur.  21rd.,  and  6fur.  ? 

SQUARE    MEASURE. 

21.  How  many  yards  are  36yd.  7ft.  126in.,  3yd.  6ft., 
130in.,  71yd.  5ft.  140in.,  and  10yd.  4ft.  21in.  ? 

22.  How  many  acres  are  34A.  3R.  32rd.,  86A.  OR 
QUd,,  381A.  2R.,  and  46A.  IR.  25rd.  ^ 


j|     10^  COMPOUND    NUMBERS.  141 

CUBIC    MEASURE. 

23,  How  much  hewn  timber  is  7T.  45ft.  1712 in., 
8T.  39ft.  1698in.,  and  lOT.  29ft.  800in..? 

24.  How  many  cords  of  wood  are  9C.  7ft. w.  15c. ft, 
4C.  6ft.  w.  12c.ft.,  and  14C.  7ft. w.  llc.ft.  ? 

fe  TIME. 

*        25.  Add  together  2Y.   250d.   ISh.   51m.   15s.,   lY. 
iSd.  7h.  Om.  55s.,  and  240d.  Oh.  37m.  29s. 

26.  Add  together  4Y.  141d.  lOh.  Om.  5s.,  12Y. 
I94d.  20h.  49m.,  and  2Y.  280d.  Oh.  55m.  38s. 

Section    10. 

COMPOUND     SUBTRACTION. 
ENGLISH    MONEY. 

1.  An  Enghsh  merchant  gave  £9176  16s.  8d.  Iqr. 
for  a  ship's  cargo,  and  then  sold  the  same  cargo  for 
JE9607  4s.  5d.  3qr.     How  much  did  he  gain  ? 

To  subtract  the  8d.,  we  unite 
1  of  the  4s.  with  the  5d.,  mak- 
ing 17d.,  and  take  8  from  17 
Then,  having  used  1  of  the  4s., 
we  unite  £1  with  3s.,  making 
23s.,  and  take  16  from  23. 

RULE  FOR  COMPOUND  SUBTRACTION.  Write  the 
several  denominations  of  the  smaller  quantity  under  the 
same  denominations  of  the  greater  quantity:  then^  begin 
with  the  lowest  denomination^  and  perform  subtraction 
on  each  denomination  separately.  Whenever  a  number 
expressing  a  denomination  in  the  upper  line  is  smaller 
than  the  number  under  it^  increase  the  upper  number  by 
as  many  as  make  1  of  the  next  higher  denomination^  and 
consider  the  number  of  the  next  higher  denomination  in 
the  upper  line^  to  be  1  less  than  it  stands. 

2.  Subtract  £4  lis.  6d.  from  £61  14s.  5d. 

3.  If  an  Enghsh  servant  receive  £  I  per  month,  and 
spend  13s.  4d.  3qr.  pcjr  month,  what  does  he  lay  up.'^ 

4.  Subtract  c£75  18s.  7d.  Iqr.  from  £856  14s.  9d  ] 


£ 
9607 
9176 

s. 

4 

16 

d. 

5 
8 

qr 
3 

1 

430 

7 

9 

2 

142  WRITTEN    ARITHMETIC.  V 

TROY    WEIGHT. 

5.  Subtract  Ub.O  oz.  19dwt.  from  21b.  lloz.  9dwt 

6.  A  silver-smith  having  41b.  3oz.  of  silver,  worked 
up  lloz.  14dwt.  of  it.     How  much  had  he  left -^ 

AVOIRDUPOIS    WEIGHT. 

7.  From  8T.  12cwt.  Iqr.  17 lb.  take  7cwt.  3qr.  21b. 
.  8.  A   farmer  laid  in  68T.  of   hay,  and  used   55T. 

I4cwt.  in  wintering  his  stock.     How  much  had  he  left  ? 
apothecaries'  weight. 

9.  From  Ift  8§  53  take  7§  73  29  16gr. 

10.  A  mixture  weighing  3g  29,  contains  89  of  jalap, 
and  the  rest  is  rhubarb.     How  much  rhubarb  ? 

CLOTH    measure. 

11.  Subtract  3qr.  2na.  from  46yd.  Iqr.  Ina. 

12.  If  7yd.  2qr.  2na.  be  cut  from  a  piece  of  cloth 
containing  46yd.  Iqr.  3na.,  how  much  will  be  left.'* 

DRY    MEASURE. 

13.  Subtract  4bu.  Ipk.  7qt.  Ipt.  from  87bu. 

14.  A  farmer  raised  lOObu.  of  corn,  and  sold  46bu. 
3pk.  of  it.     How  much  had  he  remaining .'' 

wine    MEASURE. 

15.  From  2hhd.  15gal.  take  Ihhd.  20gal.  3qt. 

16.  If  from  a  tierce  of  molasses  7gal.  2qt.  Ipt.  leak 
out,  how  much  will  remain  in  the  tierce  ? 

BEER    MEASURE. 

17.  From  4bl.  Ikil.  Ifir.  take  Ifir.  7gal.  3qt. 

18.  A  brewer  having  26bl.  Ikil.  of  beer,  sold  12bl. 
Okil.  Ifir.     How  much  had  he  remaining  ? 

LONG    MEASURE. 

19.  Subtract  4yd.  2ft.  9in.  from  5yd.  1ft.  lOin. 

20.  John  rode  16m.  5fur.,  and  Henry  rode  20m.  Ifur. 
8rd.     How  much  further  did  H.  ride,  than  J  ^ 

SQUARE     MEASURE. 

21.  A  farmer  owning  94A.  of  land,  sold  off  a  piece, 
48  rods  long,  and  20  rods  wide.  How  many  acres  had 
he  remaining  ?     (See  Square  Measure,  page  137.) 

CUBIC    MEASURE. 

22.  If  a  piece  of  timber  9  feet  long,  2  feet  wide,  and 
1  foot  thick,  be  taken  from  2T.  14ft.  of  hewn  timber, 
how  much  will  be  left.'*     (See  page  138.) 


n.  COMPOUND    NUMBERS.  143 

TIME. 

23.  Subtract  3Y.  45d.  61i.  50m.  from  5Y.  14d.  12h 

24.  A  ship  went  to  India  and  returned,  in  321  d.  7h 
How  much  less  than  a  year  was  she  in  the  voyage  ^ 

Section    II. 

COMPOUND    MULTiriJCATION. 
ENGLISH     MONEY. 

1.  What  is  the  value  of  8  yards  of  English  broad 
cloth,  at  £2  Os.  5d.  3qr.  per  yard  ? 

£     s.    d.   qr.  ^  times  3qr.  are  24qr.,  equal  to 

2     0     5     3  ^^'     ^  times  5d.  are  40d.,  and  6 

3  we  carry  are  46d.,  equal  to  3s. 

lOd.     8  times  Os.  is  Os.,  but  we 

carry  3s.     8  times  £2  are  £16. 


16     3  10     0 


RULE  FOR  COMPOUND  MULTIPIJCATION.  Begin 
uith  the  lowest  denomination^  and  multiply  each  denomi- 
nation separately ;  divide  each  product  by  the  number 
which  is  required  of  its  own  denomination  to  make  1  of 
the  next  higher;  write  the  remainder  under  the  denomi- 
nation miiltiplied^  and  carry  the  quotient  to  the  product 
of  the  next  higher  denomination. 

2.  Multiply  £529    13s.  lOd.  3qr.  by  5. 

3.  What  is  the  value  of  7  tons  of  hemp,  at  £50  18s 
lOd.  per  ton. 

4.  Multiply  £7529  ISs.  Od.  Iqr.  by  6. 

5.  Multiply  £250   16s.  lid.  by  24. 

In  examples  like  this,  it  is  most  convenient  to  multi« 
ply  hy  factors  of  the  multiplier. 

6.  Multiply  £57   8s.  lOd.  2qr.  by  45. 

7.  What  cost  34  cows,  at  £3  9s.  6d.  apiece? 
Here  find  the  price  of  32  cows  by  the  factors  of  32, 

and  to  the  product  add  the  price  of  2  cows. 

8.  Multiply  £1746   14s.  lOd.  2qr.  by  46. 

9.  What  is  the  value  of  29  yards  of  Irish  hnen,  at 
7s.  9d.  2qr.  per  yard. 

10.  Multiply  18s.  4d.  by  S3. 


144  WRITTEN    ARITHMETIC.  V 

TROY    WEIGHT. 

11.  Multiply  141b.  Ooz.  8dvvt.  llgr.  by  7. 

12.  What  is  the  weight  of  11  Federal  dollars;  the 
weight  of  1  dollar  being  ITdwt.  8gr.  ? 

AVOIRDUPOIS    WEIGHT. 

13.  Multiply  7T.  12cwt.  Iqr.  141b.  by  8. 

14.  What  is  the  weight  of  25  hogsheads  of  fish;  each 
hogshead  containing  5cwt.  3qr.  15  lb.  ? 

CLOTH    MEASURE. 

15.  Multiply  29yd.  2qt.  3na.  by  9. 

16.  How  many  yards  of  broad-cloth  are  there  in  35 
pieces;  each  piece  containing  47yd.  Iqr.  2na? 

DRY    MEASURE. 

17.  Multiply  33bu.  3pk.  6qt.  Ipt.  by  5. 

18.  How  nnany  bushels  of  corn  are  there  in  18  bags, 
each  bag  containing  2bu.  2pk.  5qt.  Ipt.  ^ 

WINE    MEASURE. 

19.  Multiply  Ip.  Ihhd.  52gal.  2qt.  Ipt.  by  4. 

20.  How   many  hogsheads  of  wine  are  there  in  13 
casks;  each  cask  containing  49gal.  3qt.  ^ 

BEER    P/IEASURE. 

21.  Multiply  6bl.  Ikil.  Ofir.  6gal.  2qt.  Ipt.  by  7. 

22.  If  1  man  drink  2gal.  3qt.  Ipt.  of  beer  in  a  week, 
how  much  will  38  men  drink  in  a  week  ? 

LONG    MEASURE. 

23.  Multiply  5lea.  2m.  6fur.  36rd.  by  8. 

24.  If  a  man  travel  55m.  5fur.   17rd.  a  day,  for  18 
days,  how  many  miles  will  he  have  travelled. 

SQUARE    MEASURE. 

25.  Multiply  36A.  3R.  27rd.  by  6. 

26.  How  many  square  yards  are  there  in  14  rolls  of 
carpeting;  each  roll  containing  52sq.  yd.  3  sq.  ft, 

CUBIC    MEASURE. 

27.  Multiply  IT.  34ft.  i200in.  of  round  timber  by  3. 

28.  There  are  4  piles  of  wood;  each  containing  3C. 
6ft. w.  12c. ft.     How  much  wood  is  there  m  all. 

TIME. 

29.  Multiply  4Y.  255d.  16h.  by  9. 

30.  If  a  ship  alter  her  latitude  1  degree  m  sailmg  16h. 
40m.,  in  what  time  will  she  alter  it  15  degrees  ? 


12  COMPOUND  NUMBERS.  |I5 

Section   12. 

COMPOUND    DIVISION. 

1.  If  £2047  13s.  9d.  be  divided  equally  among  6 
men  how  much  will  each  man  receive  ? 

We  divide  the  pounds,  and 

£         g^     ^  there  remains  <£  1 .     ThisXl 

6)2047    13    9  ^®  reduce  to  shillings,  and 

— — — — —  unite  it  with  the  13s.  making 

34  1  5  7  2qr.  333  We  divide  the  33s.,  re- 
duce the  remainder  to  pence, 
and  proceed  as  before. 

2.  If  19s.  lid.  3qr.  be  divided  equally  among  3  men, 
how  much  will  each  man  receive  ? 

3.  Divide  £16   14s.  lOd.  3qr.  equally  among  5  men. 

4.  Divide  £3  Os.  8d.  equally  among  7  men. 

5.  Divide  £59   18s.  4d.  equally  among  25  men. 

^            J     x-  1          This  operation  is  in  long 

£      s.    d.   £  s.  Q.    J.  .  .            WT     £    ..    J-  •  1 

25)59    18    4(2  7    11    ^^'^'^'««-      ^e   first    divide 

^ rr.  the  pounds:  the  quotient  is 

£2,  and  the  remamder,  £9. 

9  We  then  reduce  the  £  9  to 

^Q  shillings,  adding  in  the  18s., 

25)198(7s.  and  divide  this  sum  [198s.] 

175  as    before:    the    quotient   is 

23  7s.  and  the  remainder,  23s. 

12  We  then  reduce  the  23s.  to 

^^  pence,  adding  in  the  4d.,  and 

25)280(lld.  divide  this    sum    as   before. 

^^  5d.  remain  undivided.     Ob- 

30  serve,  that,  in  every  instance, 

25  the  quotient  and  remainder 

are  of  the  same  denomination 

with  the  dividend. 


5d.  remaining. 


RULE  FOR  COMPOUND  DIVISION.  Divide  each 
denomination  separately^  beginning  with  the  highest. 
Wlienever  a  remainder  occurs^  reduce  it  to  the  next  Zow- 
er  denomination^  add  it  to  the  number  expressed  in  the 
lower  denomination^  and  divide  it  thereunth. 

J* 


146  WRITTEN     ARITHMETIC.  V. 

6.  Divide  £460  5s.  lOd.  equally  among  37  men. 

7.  If  15cwt.  3qr.  181b.  of  flour  be  packed  equally  in 
9  barrels,  how  much  will  each  barrel  contain  ? 

8.  If  it  take  15  yards  of  cloth  to  make  6  coats,  how 
much  does  it  take  to  make  1  coat  ? 

9.  If  an  army  consume  492bu.  Opk.  6qt.  of  wheat 
in  42  days,  how  much  does  it  consume  in  1  day  .^ 

10.  Divide  3qt.  Ipt.  of  wine  equally  among  7  men. 

11.  If  30hhd.  13gal.  2qt.  of  cider  will  fill  18  casks 
of  equal  size,  how  much  does  each  cask  hold  ? 

12.  Divide  58m.  2fur.  32rd.  into  8  equal  distances. 

13.  Suppose  a  man  is  to  travel  339m.  4fur.  20rd.  in 
6  ilays;  what  distance  must  he  travel  each  day? 

14.  If  a  field  containing  22A.  2R.  12rd.  be  divided 
into  4  equal  lots,  what  will  each  lot  contain  .^^ 

15.  Suppose  a  township,  containing  17715  acR  ?  of 
land,  should  be  divided  into  80  equal  farms,  how  many 
acres  would  each  farm  contain  ? 

16.  Suppose  a  rail-way  car  to  perform  4  trips  in  5d 
16h.  9m.,  in  what  time  does  it  perform  1  trip  ? 


Q^uestions  to  be  answered  Orally. 
(1)  Which  of  these  numbers  is  a  compound  num- 
ber,—£356,  or  £2  18s.  ?  (2)  Why  is  it  called  a 
compound  number  ?  (3)  Recite  the  rule  for  addi- 
tion of  compound  numbers.  (4)  Suppose  the  sum 
of  a  column  of  numbers  expressing  furlongs  to  be  37; 
what  must  be  written  under  the  column,  and  what 
must  be  carried  to  the  next  column  ? — Why  ?  (5) 
Recite  the  rule  for  subtraction  of  compound  num- 
bers. (6)  Recite  the  rule  for  multiplication  of 
compound  numbers.  (7)  Recite  the  rule  for  divi- 
sion of  compound  numbers. 


I.  FRACTIONS.  147 

CHAP.  VI. 
FRACTIONS. 

Fractions  have  been  exemplified  in  page  44,  and  the 
mode  of  expressing  them  has  been  defined  in  page  74. 

A  proper  fraction  expresses  a  quantity,  less  than  a  unit. 
Therefore,  the  numerator  of  a  proper  fraction,  must  be 
less  than  the  denominator:  for  example,  f . 

An  improper  fraction  expresses  a  quantity,  equal  to  a 
unit,  or  greater  than  a  unit:  therefore  its  numerator  must 
equal,  or  exceed  its  denominator:  thus,  f .  |. 

A  compound  fraction  is  a  fraction  of  a  fraction — a 
part  of  a  part  of  a  unit:  for  example,  f  of  ^  . 

NOTE.  The  written  operations  requii-ed  in  the  several  sec- 
tions of  this  chapter,  con*espond  with  the  mental  operations 
involved  in  sections  of  the  same  number,  in  chapter  VI,  Oral 
Arithmetic.  Learners  will  be  enabled  to  perceive  the  written 
process  to  be  adopted,  by  recun-ing  to  the  oral  examples. 

Section    I. 

I.  What  is  the  sum  of  -^^  ^'^^  T2  ^^^  T2  - 

_3_  These  fractions  have  a  common  denom- 

5_       inator;  that  is,  they  all  have  the  same  de- 
2        nominator.     We  add  the  numerators  only, 


and  under  the  sum  of  the  numerators,  place 


12 

T2        the  common  denominator. 
2.  What  is  the  sum  of  |  and  ^  and  |  and  ^  ? 


3.  now  nmcii  is  -f^  mm  ij  anu  -fj  auu  -fj  anu  -^j  i 

4.  A  man  paid  j-^  of  a  dollar  for  breakfast,  -f^  of  a 
dollar  for  dinner,  and  -^-^  of  a  dollar  for  supper.  What 
part  of  a  dollar  did  he  spend  ? 

5.  How  much  is  23-  and  -^j  and  ^^  and  ^V  and  23-  - 

6.  A  merchant  sold  ^f  of  a  ship  to  one  man,  and  -j^  to 
another.  What  part  of  the  ship  did  he  sell } 

7.  Add  together  ^f 'and  j\  and  f|  and  jf  and  jf  ? 

8.  How  much  is  f  and  ^  and  f  and  f  ? 

9.  How  much  is  i}^  and  -^-^  and  yta  and  -^^  } 


life  WRITTEN    ARITHMETIC.  VI. 

Section  2. 

Write  upon  the  slate,  the  several  fractions  required 
in  the  following  examples. 

1.  If  you  divide  a  bushel  of  corn  into  8  equal  parts, 
and  then  put  6  of  the  parts  into  a  sack,  what  fraction  of 
a  bushel  will  there  be  in  the  sack  ? 

2.  If  an  acre  of  land  be  divided  into  20  equal  lots, 
and  14  of  the  lots  be  enclosed  by  a  fence,  what  fraction 
of  an  acre  will  there  be  in  the  enclosure  ? 

3.  Suppose  any  thing  to  be  divided  into  45  equal 
parts;  what  fraction  will  express  26  of  the  parts .'' 

4.  Suppose  1  dollar  to  be  divided  into  100  equal 
parts ;  what  fraction  will  express  one  of  the  parts  ^  2  of 
the  parts  ^     6  parts  ?     25  parts  ^     99  parts  ^ 

Section  3. 

1.  If  -^  be  subtracted  from  yj,  what  will  remain  ? 

11  Both  of  these  numerators  express  ff- 

6         teenths;  therefore  w^e  merely  subtract  one 

■ numerator  from  the  other,  and  under  the 

TT       remainder,  place  the  denominator. 

2.  If  yq  be  subtracted  from  -j^q  ,  what  will  remain.^ 

3.  What  is  the  difference  between  |-  and  f  ? 

4.  If  -^  be  subtracted  from  \j  what  will  remain  ? 

5.  What  is  the  difference  between  ^\  and  |f  ? 

6.  A  farmer  divided  a  ton  of  hay  into  20  equal  parts, 
and  gave  14  parts  to  his  cows,  and  the  rest  to  his  sheep. 
What  fraction  of  a  ton  did  the  sheep  get  ? 

7.  Subtract  ^  from  1, —  that  is,  subtract  f  from  the 
number  of  eighths  that  there  are  in  a  w^hole  one. 

8.  Subtract  ys  fi'om  a  whole  1. 

9.  What  is  the  difference  between  -j^??  and  1  ? 

10.  Subtract  |q|^  from  a  whole  1. 

11.  A  merchant  owning  a  ship,  sold  y|  of  her  to  one 
man,  ^  to  another,  and  ^\  to  another.  What  part  of 
the  ship  did  he  still  own  ? 

12.  A  boy  having  1  dollar,  paid  away  -^-^-q  of  it,  and 
lost  -^-^Q,     What  fraction  of  a  dollar  had  he  left  .^ 

13-   Subtract  f|^-  from  a  whole  I. 


2,  3.   4.   5.     RELATIONS    OF    NUMBERS.  149 

RELATIONS    OF    NUMBERS. 

Section  4. 

We  frequently  have  occasion  to  view  one  number  as 
a  certain  part  of  another  number;  and  thus  we  notice 
the  relation  that  exists  between  the  two  numbers.  In 
order  to  state  what  part  one  number  is,  of  another,  we 
m«ke  the  number  which  is  the  part  a  numerator,  and  the 
other  number  a  denominator. 

State  the  fractions,  which  answer  to  the  following  ex- 
amples, upon  the  slate. 

1 .  What  part  of  5  cents  is  1  cent  ?     is  3  cents  ? 

2.  What  part  of  10  is  1  ?     is  2  ?     is  5  ?     is  9  ? 

3.  What  part  of  £1  or  20  shillings,  is  1  shilling.^  is 
6  shillings  ?     is  14  shilhngs  ? 

4.  What  part  of  35  is  1  ?     is  8  ?     is  11  ?     is  34  ? 

5.  What  part  of  $  1  or  100  cents,  is  1  cent?  is  2 
cents  ?     is  9  cents  ?     is  46  cents  ?     is  94  cents  ? 

6.  What  part  of  6  pence  is  1  penny  ?     is  5  pence  } 

7.  What  part  of  1  shilling  is  1  penny  ?     is  7  pence  } 

8.  W^hat  part  of  1  peck  is  1  quart  ?     is  7  quarts  ? 

9.  What  part  of  1  hogshead  is  1  gallon  }     is  18  gals,  r 

10.  If  ^V  ^^  ^  hogshead  of  wine  be  worth  $1,  what 
is  -5^  of  a  hhd.  worth  ?     What  is  Ihhd.  worth  ? 

11.  What  part  of  1  year  is  1  day  ?  is  10  days  .'^  is  40 
days  ?     is  100  days  ?     is  275  days  ? 

12.  If  a  man  spend  $  1,  in  3-^-5  of  a  year,  how  much 
will  he  spend  in  -^-^  of  a  year  ?  in  ^|f  of  a  year  } 
How  much  will  he  spend  in  1  year  ? 

13.  What  part  of  2016  is  1  ^     is  84  }     is  759  .? 

Section  5. 

1.  Suppose  ^  of  a  ship  to  be  worth  $4703;  what  is 
the  whole  ship  worth  .^ 

2.  4703  is  ^  of  what  number } 

3.  If  \  of  an  acre  of  land  produce  71  bushels  of  po- 
tatoes, how  many  bushels  will  1  acre  produce  ? 

4.  71  is  -3-  of  what  number  } 

5.  875  is  4-  of  what  number  ? 

6.  1900  is  5  of  what  number  .'^ 

N* 


iiSO  WRITTEN    ARITHMETia  VI 

7.  If  230  men  will  lay  -^  of  a  mile  of  rail-way  a  week, 
how  many  men  will  it  take  to  lay  1  mile  in  a  week  t 

8.  230  is  \  of  what  number  ? 

9.  44  is  Y  of  what  number  ? 

10.  6902  is  \  of  what  number  ? 

11.  li-ii  of  a  pound  of  silver  be  worth  $1.09,  what 
is  1  pound  of  silver  worth  ? 

12.  If  a  ship  sail  17  miles  in  ^V  of  a  day,  what  dis- 
tance would  she  sail  in  the  whole  day  ? 

13.  204  is  3^  of  what  number  ? 

14.  If  "x^o  ^^^  P^P^  of  wine  be  worth  $1.15,  what  is 
the  whole  pipe  of  wine  worth  ? 

15.  Suppose  \-  of  the  sugar  in  a  hogshead  to  weigh 
Icwt.  2qr.  121b.;  what  does  the  whole  weigh? 

Section  6. 

1.  If  1  acre  of  land  will  produce  126  bushels  of  po- 
tatoes, how  many  bushels  will  \  of  an  acre  produce  } 

2.  What  is  -J  of  126  ? 

3.  Suppose  38406  needles  can  be  made  from  a  bar 
of  steel;  how  many  can  be  made  from  \  of  the  bar  ? 

4.  What  is  -^  of  38406  ? 

5.  If  1  dollar  will  pay  for  316  quills,  what  number  of 
quills  will  '\  of  a  dollar  pay  for  ? 

6.  If  you  eat  1095  meals  in  1  year,  what  number  of 
meals  do  you  eat  in  ^  of  a  year  ? 

7.  What  number  of  cubic  inches  are  there  in  -g^  of  a 
cubic  foot?     (See  Cubic  Measure,  page  132.) 

8.  If  1  week's  board  cost  $3.64,  what  does  ^  of  a 
week's  board  cost  ? 

9.  Suppose  a  packet  ship  to  be  worth  $17841.50; 
what  is  -iQ  of  her  worth  ? 

10.  A  man,  having  $205.12,  paid  -j^  ^^  his  money 
for  a  piece  of  land.     What  was  the  price  of  the  land  ? 

11.  A  man  gave  $2568  for  a  house,  and  then  paid  ^i 
part  as  much  for  having  it  repaired.  For  how  much 
must  he  sell  the  house,  in  order  to  lose  nothing  ? 

12.  What  is  yV  of  1800? 

13.  Suppose  a  piece  of  cloth  to  contain  60yd.  2qr. 
how  much  cloth  is  there  in  \  of  the  piece  ? 


3    7.  8.  RELATIONS    OF    NUMBERS.  I5l 

Section  7. 

1.  Suppose  that  12  men  are  to  pay  a  debt  of  $420, 
iti  equal  shares;  what  must  1  man  pay? 

Solution.  1  man  is  j2  ^^  ^^  men;  therefore  1  man 
must  pay  -^^  of  $420.     -^~  of  420  is  — 

2.  If  a  prize  of  $3936  be  divided  equally  among  8 
men,  what  part  of  the  money  will  1  man  receive  ?  How 
many  dollars  will  1  man  receive  ? 

3.  27  men  own  864  acres  of  land  together.  What 
part  of  864  acres  does  1  man  own  ?  What  number  of 
acres  does  1  man  own  ? 

4.  If  $  135.45  will  pay  for  Ihhd.  of  wine,  what  part 
of  the  money  would  pay  for  1  gallon  ?  What  would  be 
the  price  of  1  gallon  ? 

5.  If  170  acres  of  land  produce  6630  bushels  of  corn, 
what  part  of  6630  bushels  does  1  acre  produce  ?  How 
many  bushels  does  1  acre  produce  ? 

6.  If  6  yards  of  broad-cloth  be  worth  J£ll  lis.  9d., 
what  part  of  the  money  is  1  yard  worth  ?  What  is  the 
value  of  1  yard,  in  pounds,  shillings,  &c. } 

7.  A  black-smith  paid  $  63  for  1 5  tons  of  coal.  What 
did  the  coal  cost  him  per  ton  ? 

Section  8. 

1.  A  man  purchased  a  farm  for  $5642,  and  paid  j  of 
the  price  in  cash,  and  gave  his  note  for  the  remainder. 
How  many  dollars  did  he  pay  down  ? 

Direction,  First  find  y  of  $5642,  by  dividing  this 
«um  by  the  denominator  of  the  fraction;  then  find  4- 
sevenths,  by  multiplying  the  quotient  by  the  numerator. 

2.  Whatisf  of  1905? 

3.  If  an  acre  of  land  will  produce  14870  ears  of  corn, 
how  many  ears  will  f  of  an  acre  produce  ? 

4.  What  is  I  of  19064  ? 

5.  Suppose  an  acre  of  land  to  be  worth  $48.16;  what 
is  the  value  of  |  of  an  acre  of  the  same  land  ? 

6.  If  1  dollar  will  pay  for  270  quills,  what  number  of 
quills  will  j^Q  of  a  dollar  pay  for  ? 

7.  If  72  gallons  of  wine  leak  from  a  pipe  in  1  dav^ 
now  many  gallons  leak  out  in  ^  of  a  day  ? 


152  WRITTEN   ARITHMETIC.  VI. 

8.  Suppose  a  hogshead  of  sugar  to  be  worth  J£20  4s. 
3d. ;  what  is  the  value  of  ^  of  the  sugar  ? 

9.  What  is  if  of  4720  ? 

In  the  several  foregoing  examples  in  this  section,  the 
earner  has  probably  divided  the  given  number  by  the 
denominator  of  the  fraction,  and  multipHed  the  quotient 
Dy  the  numerator.  It  is,  however,  sometimes  more 
convenient,  to  multiply  the  given  number  by  the  nume- 
rator, and  divide  the  product  by  the  denominator. 

10.  What  is  I  of  32  .?     (Here  are  the  two  methods.) 


4)32 

8 
3 

First    Method, 
""is  i  of  32. 

Second    Method. 
32 
3 

4)96    is  3  times  32. 

24 

is  3  times  |  of  32, 
which  is  1  of  32. 

24    is  1  of  3  times  32, 
which  is  1  of  32. 

We  may  see  why  these  two  methods  of  operation 
produce  the  same  result,  in  the  following  illustration. 
Here  is  |  of  32  units  arranged  in  one  line,  and  \  oi  S 
times  32  units  arranged  in  three  lines.  The  number  of 
units  [®]  in  the  two  arrangements  is  the  same. 

a®®®o®®®  ^®oeoe®o  ®#oe9oe®  oooooooo 

©®®®®©®®  oooooooo  oooooooo  oooooooo 
•••«<»«©«  oooooooo  oooooooo  oooooooo 
•••®<^©®®  oooooooo  oooooooo  oooooooo 

11.  Find  -/tj  of  60156,  by  each  of  the  above  methods. 

12.  Find  sf y  of  10849,  by  the  second  method. 

13.  A  laborer  worked  |  of  a  year,  at  92  cents  per 
day.     What  did  his  wages  amount  to? 

14.  In  \^  of  a  pipe  of  wine,  how  many  gallons? 

15.  What  is  1-^0  of  $1491? 

After  multiplying  by  6  and  dividing  by  100,  reduce 
the  remainder  to  cents,  and  divide  the  cents. 

16.  A  borrowed  of  B,  $  758,  promising  to  pay  it  in 
one  year;  and,  in  addition  thereto,  he  agreed  to  pay  a 
sum,  equal  to  yf  o  ^^  ^^^  ^"^^  borrowed,  for  the  use  of 
the  money.     How  much  must  B  receive? 

17.  Whatisy^o  of$2S? 


9.   10.  RELATIONS    OF    NUMBERS.  1W 

Section  9. 

1.  If  $686.56  should  be  divided  equally  among  8 
men,  what  part  of  the  money, — and  what  number  of 
dollars  and  cents,  would  3  men  receive  ? 

2.  Suppose  that  $33  will  pay  for  198  yards  of  cloth; 
what  part  of  198  yards, —  and  how  many  yards  can  be 
bought  for  $  14  ? 

3.  If  15  acres  of  land  produce  283bu.  Opk.  4qt.  of 
wheat,  what  part  of  this  quantity, —  and  how  many  bush- 
els will  9  acres  produce  ? 

4.  If  540  barrels  of  flour  will  supply  an  army  for  30 
days,  how  many  bushels  will  supply  it  for  19  days  ? 

Solution.  19  days  are  ^-|  of  30  days;  therefore  the 
army  will  consume  ^^  of  540  barrels. 

5.  If  a  man  can  build  256  rods  of  fence,  in  60  days, 
how  many  rods  can  he  build  in  45  days  ? 

6.  If  72  tons  of  hemp  cost  $  13680,  what  will  20  tons 
cost,  at  the  same  rate  ? 

7.  If  it  take  a  man  31  days  to  travel  1178  miles,  how 
many  miles  can  he  travel  in  25  days  ? 

8.  If  24  English  watches  are  worth  ^6108  18s. ,  what 
IS  the  value  of  7  watches  of  the  same  kind  ? 

Section   10. 

1.  If  16  rilen  can  fell  208  trees  in  a  day,  how  many 
trees  can  35  men  fell  in  the  same  time  ? 

2.  What  is  35  times  ^V  of  ^^^  ? 

3.  If  10  barrels  of  flour  cost  $59.30,  how  much  will 
33  barrels  cost,  at  the  same  price  per  barrel  ? 

4.  What  is  33  times  iV  of  $59.30  ? 

5.  If  64  soldiers  eat  448  pounds  of  beef  in  a  week, 
how  many  pounds  will  250  soldiers  eat  in  a  week  ? 

6.  What  is  250  times  -J4  of  448  } 

7.  If  12  gallons  of  linseed  oil  be  sold  for  $13.44, 
what  should  be  the  price  of  52  gallons  of  linseed  oil  ? 

8.  What  is  52  times  -^^  of  $  13.44  ? 

9.  If  a  man  earn  $91.70  in  7  months,  how  much  can 
he  earn  in  2  years  ? 

10.  If  48  pounds  of  feathers  can  be  bought  for  $16, 
how  many  pounds  can  be  bought  for  $25  ? 


154  WRITTEN    ARITHMETIC.  VI 

Section    11. 

1.  If  f  of  an  acre  of  land  will  produce  28  bushels  of 
potatoes,  how  many  bushels  will  ^  of  an  acre  produce  ? 
How  many  bushels  will  1  acre  produce  ? 

2.  If  I  of  a  hogshead  of  sugar  be  worth  $22.50,  what 
s  ^  of  it  worth  ?     What  is  the  whole  worth  ? 

3.  $22.50  is  I  of  what  sum  of  money  ? 

4.  Suppose  a  ship  to  sail  105  miles  in  -^2  of  a  day; 
what  distance  will  the  ship  sail  in  -^^  ^^  ^  ^^Y  -  What 
distance  will  she  sail  in  the  whole  day  ? 

5.  105  is  y2  ^f  ^^hsit  number? 

6.  If  -^j  of  a  chest  of  tea  be  worth  $23,  what  is  yV 
of  it  worth  ?     What  is  the  whole  of  it  worth  ? 

7.  $23  is  lY  of  what  sum  of  money? 

8.  If  f  of  a  bag  of  coffee  be  worth  $38.46,  what  is 
the  whole  bag  of  coffee  worth  ? 

9.  Suppose  a  rail-way  car  to  run  198  miles,  in  -^-^  of 
a  day;  what  distance  will  it  run  in  1  day  ? 

10.  If  192  men  will  perform  f  of  a  certain  piece  of 
work  in  a  week,  what  number  ojf  men  will  it  take,  to 
perform  the  whole  of  the  work  in  a  week  ? 

11.  192  is  f  of  what  number  ? 

12.  A  man  purchased  a  farm,  and  after  he  had  paid 
^  of  the  price,  he  still  owed  $1288.  What  must  have 
been  the  price  of  the  farm  ? 

13.  A  trader  purchased  a  pipe  of  wine,  and  after  ^  of 
it  had  leaked  out,  he  sold  the  remainder  at  $1.15  per 
gallon.     How  much  did  it  amount  to  ? 


Questions  to  be  answered  Orally. 
(1)  What  is  meant  by  a  common  denominator  of 
two  or  more  fractions  ?  (2)  How  do  you  add  frac- 
tions, that  have  a  common  denominator  ?  (3)  How 
do  you  subtract  one  fraction  from  another;  the  two 
fractions  having  a  common  denominator  ?  (4) 
When  a  certain  fractional  part  of  a  number  is  known, 
how  do  you  find  the  whole  of  the  number?  (5) 
When  the  whole  of  a  number  is  known,  how  do  you 
find  any  certain  fractional  part  of  it  ? 


11.    12.    13.    RELATIONS     OP    NUMBERS.  155 

Section    12. 

REVIEW. 

1.  (§1.)  If  you  should  pay  -jYo  ^^  ^  dollar  for  a  quire 
of  paper,  -^-^^  of  a  dollar  for  a  slate,  and  ^q^q  for  a  book, 
what  fraction  of  a  dollar  would  you  spend  ? 

2.  (§2.)  If  a  ton  of  hay  be  rolled  uy  in  20  equal 
heaps,  what  fraction  of  a  ton  will  14  heaps  be  ? 

3.  (§  3.)  Suppose  a  young  man  to  lay  out  \^  of  his 
money  for  a  farm;  what  part  of  his  money  has  he  left  ? 

4.  (§  4.)  Suppose  a  school  to  be  allowed  15  minutes 
for  recess;  what  fraction  of  an  hour  is  the  recess  ? 

5.  (§  5.)  2V  ^f  ^"  ounce,  or  Idwt.  of  pure  gold,  is 
sufficient  ^to  gild  a  silver  wire,  65  miles  in  length  What 
length  of  wire  may  be  gilded  with  1  ounce  ? 

6.  (§6.)  If  a  man's  income  be  ^^193  a  year,  how 
much  is  his  income  for  ^2  of  a  year,  or  1  month  ? 

7.  ( §  7.)  If  $  14  will  pay  for  70  books,  what  part  of 
70  books,—  and  how  many  books,  will  ^  1  buy  ^ 

8.  (§8.)  If  a  man's  income  be  ^803  a  year,  how 
much  is  his  income  for  3W  of  a  year,  or  30  days  ? 

9.  (§9.)  Suppose  $93.66  to  be  paid  for  14  yards 
of  broad-cloth;  what  part  of  the  money  does  6  yards 
cost?     How  many  dollars  do  6  yards  cost.'^ 

10.  (§  10.)  If  11  barrels  of  flour  are  worth  $59.07, 
what  is  the  value  of  25  barrels,  at  the  same  rate  ? 

11.  (§11.)  If  18  shillings  be  given  for  --^^  of  a  hun- 
dredweight of  fish,  what  must  be  given  for  1  cwt.  ? 

FRACTIONS  AND  RELATIONS. 

Section    13. 
1.  Suppose  you  can  read  12  pages  in  an  hour;  how 
many  hours  will  it  take  you  to  read  160  pages  ? 

12)160(13-^-  We  find,  by  division,  it  will  take 

12  13  hours,  and  still  4  pages  remain 

""TTT  to  be  read.     Now,  since  it  takes 

OQ  1^2  ^f  ^"  ^^^"^  ^^  ^'^^^  ^  P^g®?  ^t 

will  take  1^2  ^^  ^"  ^^^^  ^^  ^'^^^  ^ 

4  pages.     Ans.  13^^  hours. 


156  WRITTEN    ARITHMETIC.  VI 

Jlny  REMJilKDER,  which  appears  after  the  operation  of 
division^  is  the  numerator  of  a  fraction^  the  divisor  being 
the  denominator;  and  this  fraction  forms  a  part  of  the 
quotient.  Therefore,  place  the  remainder  and  the  divi' 
sor^  as  a  fraction^  to  the  right  of  the  quotient, 

2.  At  |i2 per  yard,  how  many  yards  of  cloth  can  be 
bought  for  ^49;  that  is,  how  many  whole  yards,  and 
what  part  of  another  yard  can  be  bought  ? 

3.  How  many  times  2  are  there  in  49  ? 

4.  How  much  flour  can  be  bought  for  ^639,  at  $5 
per  barrel;  that  is,  how  many  barrels,  and  what  part  of 
another  barrel  can  be  bought  ? 

5.  In  639,  how  many  times  5; — that  is,  how  many 
fives,  and  what  part  of  another  5,  in  639  ? 

6.  How  much  salt  can  be  bought  for  87  shillings,  at 
4  shillings  per  bushel  ? 

7.  Suppose  a  rail-way  car  to  run  16  miles  an  hour' 
in  how  many  hours  will  it  run  350  miles  ? 

8.  How  many  times  12  are  there  in  1049  ? 

9.  How  many  times  39  are  there  in  76800  ? 

10.  In  438  shillings,  how  many  pounds  are  there  ? 
Observe  that  Is.  is  gV  of  "^l?  therefore  the  remain- 
der in  this  example,  may  be  expressed  as  a  fraction. 

11.  How  many  yards  of  cloth  can  be  bought  for  549 
shillings,  at  <£1  per  yard? 

12.  Suppose  a  ton  of  hay  to  be  equal  in  value  to  34 
bushels  of  oats;  how  many  tons  of  hay  must  be  given 
for  450  bushels  of  oats  ? 

13.  How  many  times  17  are  there  in  23  times  31  ? 

14.  How  much  rice,  at  $4  per  cwt.,  must  be  given 
for  6201b.  of  cheese,  at  10  cents  per  pound  ? 

Section    14. 

CHANGE  OF  WHOLE  NUMBERS  TO  FRACTIONS. 

1.  How  many  thirds  are  there  in  14.^ 

.  .  In   1   there  are  3-thirds, 

o  therefore,  there  are  3  times 

^  as  many  thirds  as  whole  ones, 

42-thirds.    *Ans.  3  .    in  any  whole  number. 


14  15.    FRACTIONS  AND  RELATIONS.       157 

RULE.  To  change  a  whole  number  to  an  improper 
fraction.,  multiply  the  whole  number  by  the  denominator^ 
and  the  product  will  be  the  numerator. 

2.  If  you  cut  17  sheets  of  paper  into  half  sheets,  how 
many  halves  will  there  be  } 

3.  How  many  fifths  of  a  dollar  are  there  $  IG  .^ 

4.  In  31  pounds,  how  many  sixths  of  a  pound  ? 

5.  In  73  yards,  how  many  eighths  of  a  yard  } 

6.  Change  641  to  ninths.     Change  641  to  tenths. 

7.  If  a  stage  run  1  mile  in  \  of  an  hour,  how  many 
miles  would  it  run  in  126  hours  ^ 

'  8.   How  many  fourths  are  there  in  15-|.^ 

j^_3  In  this  example,  we  add  the 

A^  3-fourths  to  the  fourths  produc- 

• ed  by  the  multiplication  of  15 

6  3-fourths.  i^^  4^  ^^^^  ^^^^  ^l^^^j^  6_3  ^ 

J^ote.  A  whole  number  and  a  fraction  expressed  to- 
gether, thus,  15f ,  is  called  u.  mixed  number. 

9.  How  many  eighths  of  a  mile  in  57^  miles  "^ 

10.  Change  Q6\y  ^o  an  improper  fraction. 

11.  Change  4-^^  to  an  improper  fraction. 

12.  If  3^  of  a  dollar  will  pay  for  1  gallon  of  beer,  how 
much  beer  can  be  bought  for  $6^^ 

13.  If  |-  of  a  dollar  will  pay  for  3  yards  of  ribbon, 
how  many  yards  can  be  bought  for  $6^? 

Section    15. 

CHANGE  OF  FRACTIONS  TO  WHOLE  NUMBERS. 

1.  How  many  whole  ones  are  there  in  "^-^  ? 

3)42-thirds.  S^^^^^  f  make  a  whole  1,  there 

—  are  as  many  whole  ones  m  y 

14  wholes.         j^g  ^YiQ^,Q  ^^^  tinies  3  in  42. 

RULE.  To  change  an  improper  fraction  to  a  whole 
number^  divide  the  numerator  by  the  denominator^  and 
the  quotient  will  be  the  whole  number. 

2.  How  many  whole  sheets  of  paper  must  be  cut  into 
halves,  to  make  "^^  of  a  sheet  ^ 


i5S  WRITTEN     ARITHMETIC.  VI. 

'    3.  In  ^1^  of  a  dollar,  how  many  dollars  ? 

4.  How  many  pounds  ai-e  there  in  J-|^  of  a  pound  ? 

5.  In  ^-%-^  of  a  yard,  how  many  yards  are  there  ? 

6.  If  a  stage  run  1  mile  in  ^  of  an  hour,  how  many 
hours  would  it  be  in  running  126  miles  ? 

7.  Change  ^4^  to  a  whole  number. 

8.  Suppose  8  pounds  of  sugar  can  be  bought  for  $1; 
how  many  pounds  will  -ff-  of  a  dollar  pay  for  ? 

9.  Change  ^/  to  a  mixed  number  ? 

rNgy  We  divide  the  S7-fifths  by  5,  and 

^ obtain  17  whole  ones:  then,  there  are 

I'^f  2-fifths  over,  making  17-|. 

10.  Change  -f-  to  a  mixed  number. 

11.  How  many  dollars  are  there  in  -^f^  of  a  dollar.^ 

12.  How  many  gallons  in  ''-f^  of  a  gallon  ? 

13.  If  -5-  of  a  dollar  will  pay  for  1   pound  of  coffee, 
how  many  dollars  wdl  312  pounds  of  coffee  cost  ? 

14.  If  1  pound  of  butter  cost  \  of  a  dollar,  what  would 
be  the  price  of  491  pounds,  at  the  same  rate  ? 

Section   16. 

The  sum  of  these  fractions  will  be  an  improper  frac- 
tion, and  it  must  be  changed  to  a  mixed  number. 
2.  What  is  the  sum  of  491 -i\,  75fx  and  SSG^^y  ? 
49]_9..  In  this  example,  we  add  together 

7  5_B_  the  elevenths,  and  find  their  sum  to 

g3^_b  be  ff;  which  is  equal  to  2-i\.       The 

-^1  we  write  down,  and  carry  the  2  to 


1 404-^-  11  c 

x-t\j-x^-^  the  column  01  units. 

3.  What  is  the  sum  of  419f,  ISf,  12f ,  8573f ,  9j, 
251  f,  141-1,  and  25-|? 

4.  Add  together  336^^5,  14i-|,  970Uf,  28,  158^-?-, 
1240^5,  l\,  lOO^V^  and  -i|. 

6.   Subtract  1876|  from  2258|. 

2  0  c  Q  3    •         We  cannot  take  |  from  ^ ,  therefore, 
1075!         we  join  1  unit  with  the  |-,  making  ^/, 
**         and  take  |  from  ^f  .    We  then  proceed 
to  take  6  units  from  7  units. 


'  9 


16.  17.  18.   FRACTIONS     AND    RELATIONS.  15& 

6.  Subtract  46031|  from  71706}. 

7.  Subtract  609 |i-  from  5420067^. 

8.  If  17|  yards  of  cloth  be  cut  from  a  piece  contain- 
ing 49  yards,  how  much  will  be  left  ? 

9.  A  retailer  put  into  a  firkin,  28-]^^  pounds  of  butter 
at  one  time,  19^  pounds  at  another,  and  SSyg-  pounds 
at  another,  and  then  sold  out  25 ^|  pounds.  How  many 
pounds  still  remained  in  the  firkin  ? 

Section   17. 

1.  Suppose  a  rail-road  car  to  run  f  of  a  mile  in  1 
minute;  what  distance  will  it  run  in  47  minutes  ? 

2.  How  many  whole  ones  in  47  times  f  ? 

3.  If  ^  of  a  yard  of  broad-cloth  will  make  1  jacket, 
how  many  yards  will  it  take  to  make  18  jackets  ? 

4.  How  many  whole  ones  in  18  times  |-? 

5.  If  I  of  a  pound  of  gunpowder  tea  cost  1  dollar, 
how  many  pounds  can  be  bought  for  50  dollars  ? 

6.  How  many  whole  ones  in  50  times  ^} 

Section   18. 
I     1.  What  is  the  product  of  86f ,  multipHed  by  9? 

QQ4  We  multiply  f  by  9  thus,  9  times  y 

g^         is  ^j  ;  eq^al  to  5-^.     Then  we  write 

this  Y  under  the  |,  and  carry  the  5  to 

'^'^^T         the  product  of  the  6  units. 

2.  What  is  the  product  of  41|,  multiplied  by  7  ? 

3.  What  is  the  value  of  a  field,  which  contains  5 
acres,  allowing  it  to  be  worth  54^  dollars  per  acre  .-^ 

4.  How  much  is  4  times  35 ^§  ? 

5.  How  much  is  9  times  14731 1|? 

6.  How  much  is  28  times  54  f? 

54f  Here  we  are  obhged  to  multiply  by 

28  8  units  and  2  tens  separately,  and  we 

432  cannot  well  bring  in  the  product  of  the 

108  fraction   by    the    2  tens.     Therefoce, 

132  we  first  multiply  the  whole  numbers, 

Y"  — -J  and  then  find  28  times  |,  m  a  separate 

^oc^K)^  operation,  which  is  not  here  written. 


160  WRITTEN     ARITHMETIC.  VI 

7.  How  much  is  92  times  2051 -j^q  ? 

8.  How  much  is  100  times  M^j? 

9.  How  many  gallons  of  wine  are  there  in  31  casks, 
each  cask  containing  54 -j^^  gallons  ? 

10.  A  merchant  paid  $75 -j^^  apiece,  for  47  mules. 
What  did  the  whole  amount  to  ? 

11.  If  a  steam-boat  run  236^  miles  in  1  day,  what 
distance  will  it  run  in  16  days  ? 

12.  How  much  is  15  times  f^? 

13.  What  cost  75  books,  at  -jYo  ^^  ^  dollar  apiece  } 

14.  If  a  horse  eat  -||  of  a  bushel  of  oats  a  day,  how 
many  bushels  will  he  eat  in  365  days  ? 

Section  19. 

1.  If  I  of  a  chest  of  tea  be  worth  $6.87^,  what  is 
the  whole  chest  worth  ? 

2.  If  j\^  of  a  dollar  will  pay  for  travelling  27|  miles 
on  a  turnpike  road,  how  far  can  you  go  for  $i? 

3.  790 1  is  jV  ^f  ^vhat  number  ? 

4.  If  4.  of  a  kite  line  be  25 1  yards  in  length,  what  is 
the  whole  length  of  the  line  ? 

5.  305-/3  is  ^  of  what  number  ? 

6.  If  a  man.can  earn  12^  cents  in  ^  of  a  day,  what 
sum  of  money  can  he  earn  in  1  day  ? 

7.  If  ^  of  a  yard  of  gold  wire  be  worth  y|  of  a  dollar, 
what  is  the  value  of  1  yard  of  the  wire  ? 

Section  20. 

1.  A  boy  having  $2,  gave  |  of  his  money  for  a  knife. 
What  fraction  of  1  dollar  did  the  knife  cost  ? 

2.  -5-  of  2  is  equal  to  what  part  of  1  ? 

3.  6  men  divided  5  barrels  of  flour  equally  among 
them,  each  man  taking  ^  of  the  flour  in  each  barrel 
What  fraction  of  a  barrel  did  each  man  get  ? 

4.  "3^  of  5  is  equal  to  what  part  of  1  ? 

5.  If  you  should  take  ^-^  of  a  bushel  of  corn  from  each 
of  10  bushels,  what  fraction  of  1  bushel  would  you  obtain  ? 

6.  What  part  of  1  is  ^j  of  10  ? 

7.  What  part  of  1  is  3V  of  2  ?  is  ^  of  3 }  is  ^\o{ 4  ? 
is -sV  of  18?     is  3V  of  38.^ 


I 


19.  20.21.  FRACTIONS  AND  RELATIONS.       16! 

8  A  tenant  raised  28  bushels  of  corn,  and  gav^e  his 
landlord  ^  of  it.  What  improper  fraction  of  a  bushel, 
[how  many  thirds  of  a  bushel,]  did  the  landlord  receive  ^ 
How  many  bushels  did  the  landlord  receive  ? 

9.  -^  of  23  is  equal  to  what  improper  fraction  ?  Then 
^  of  28  is  equal  to  how  many  whole  ones  ? 

10.  I  of  42  is  equal  to  what  improper  fraction  ?  Then 
5  of  42  is  equal  to  what  mixed  number  ? 

11.  -5^3  of  29  is  equal  to  what  improper  fraction.'* 
Then  -^^3  of  29  is  equal  to  what  mixed  number  ? 

12.  If  $721  should  be  divided  equally  among  6  men, 
how  many  sixths  of  a  dollar  would  each  man  have  ? 
How  many  dollars  would  each  man  have  ^ 

Section   21. 

1.  Suppose  a  hogshead  of  brown  sugar  to  be  worth 
^115;  what  is  the  value  of  -}  of  the  sugar.'* 

2.  What  is  -}of  115? 

8.  5  men  from  Connecticut,  bought  793  acres  of  land 
in  Michigan,  and  divided  it  into  5  equal  farms.  How 
many  acres  were  there  in  each  farm  ? 

To  find  J  of  793  acres,  we  divide  793  by  5.  The 
quotient  is  158  acres,  and  there  is  a  remainder  of  3  acres. 
To  divide  these  3  acres,  we  take  -\  of  each  acre  for  each 
farm.     -]-  of  3  acres  is  f  of  1  acre. 

4.  What  is  ■}  of  1315  ?     -J-  of  530  ?     -J  of  8201  ? 

5.  Suppose  12  men  to  share  equally  in  a  prize  of 
$551.20;  what  is  each  man's  share  ? 

G.  What  is  1^5  of  55120  ?     ^V  ^f  967  .?     ^V  of  700  > 

7.  The  Young  Ladies''  Class  Book^  which  consists 
of  408  pages  of  select  reading  lessons,  has  been  read 
through  by  25  scholars;  each  reading  an  equal  portion. 
How  many  pages  did  1  scholar  read  ? 

8.  \  of  o€l  \\  of  20s.]  is  how  many  shillings,  and 
what  fraction  of  a  shilling  ? 

9.  Find  \  of  £7  in  shillings  —  that  is,  reduce  J£7  to 
Bhillings,  and  find  \  of  the  number  of  shillings. 

10.  -g^  of  5  shillings  is  how  many  pence  ? 

11.  How  many  grains  in  -^^  of  G  pennyweights  ^ 

12.  -^  of  3  ounces  is  how  many  drams  ^ 


162  WRITTEN    ARITHMETIC.  VI, 

13.  11  men  divided  9  hogsheads  of  molasses  equally 
among  them.     How  many  gallons  had  each  man  i 

14.  Y5  of  7  furlongs  is  how  many  rods .'' 

15.  Y  of  6  square  feet  is  how  many  square  inches  ? 

16.  How  many  seconds  are  there  in  -^^  ^f  ^^  hours? 

17.  A  trader  sold  -4-  of  a  hogshead  of  wine  at  37  cents 
for  every  -^  of  a  gallon.     What  did  it  amount  to  ? 

Section    22. 

1.  If  34  barrels  of  flour  be  made  from  147  bushels  of 
wheat,  how  much  wheat  will  make  9  barrels  of  flour  ? 

2.  What  is  9  times  -^-^  of  147  ? 

3.  If  8  yards  of  broad-cloth  cost  ^^38,  what  will  13 
yards  cost,  at  the  same  rate  ? 

4.  Wh^t  is  13  times  ^  of  38  ?  7  times  ^V  ^f  ^^4  } 
5  times  jV  of  270  ^     21  times  iV  of  40975  ? 

5.  If  it  cost  $1.25  to  ride  20  miles  in  a  stage,  how 
much  will  it  cost  to  ride  32  miles  ^ 

6.  If  ^  16  will  pay  for  85  pounds  of  butter,  how  many 
pounds  will  $25  pay  for  ? 

7.  If  12  cords  of  wood  cost  $75,  what  will  be  the 
cost  of  19  cords,  at  the  same  rate  .'^ 

8.  Suppose  21cwt.  of  flour  to  be  worth  the  same  as 
65  bushels  of  salt;  how  many  bushels  of  salt  must  be 
given  in  exchange  for  IScwt.  of  flour.'* 

9.  If  8  barrels  will  hold  19  bu.  3pk.  4qt.  of  corn,  how 
much  corn  can  be  put  into  15  barrels  ? 

Section   23. 

1.  When  writing  paper  is  sold  at  $5.42  per  ream, 
what  is  the  price  «f  -]-  of  a  ream  ? 

2.  If  ^  of  a  ream  of  paper  is  worth  $  1.35| ,  what  is 
I  of  a  ream  worth  ? 

3.  Suppose  a  hogshead  of  sugar  to  be  worth  $93; 
what  is  -^  of  it  worth  ? 

4.  If  I  of  a  hogshead  of  sugar  is  worth  $13,284, 
what  is  4  of  it  worth  ? 

5.  Wiiat  is  i  of  .3765  ?     What  is  f  of  3765  ? 

6.  How  many  gallons  are  there  in  ^  of  a  pipe  of  wine  f 
How  many  gallgns  ia  ^  of  a  pipe  ? 


e2.  23.  PERCENTAGE.  163 

In  the  following  examples,  it  will  be  most  convenient 
to  multiply  by  the  numerator  of  the  fraction,  before 
dividing  by  the  denominator.  See  Second  Method  of 
operation,  exemplified  in  page  152. 

7.  The  highest  point  of  the  Andes,  is  21440  feet: 
Mont  Blanc,  of  the  Alps,  is  |-  as  high.  What  is  the 
height  of  Mont  Blanc  } 

8.  Virginia  contains  06000  square  miles:  Rhode 
Island  is  only  ^  as  large.  How  many  square  miles  are 
there. in  Rhode  Island  ? 

9.  If  a  man's  income  be  $  1000  per  year,  what  is  his 
income  for  9  months,  or  -{^  of  a  year  } 

10.  Suppose  a  man  by  constant  industry  can  earn 
$1.50  per  day;  what  will  he  earn  in  10  days,  allowing 
him  to  rest  f  of  the  time } 

11.  How  much  must  be  paid  for  \^  of  a  ton  of  Russia 
hemp,  when  the  price  is  $210  per  ton } 

12.  A  man  having  4  miles  to  go,  rode  ^  of  the  way, 
and  walked  the  remainder.     How  many  rods  did  he  walk  } 

13.  Suppose  45000  pounds  of  iron  to  be  sufficient  to 
lay  the  track  on  1  mile  of  rail-way;  how  many  pounds 
of  iron  are  required  to  lay  the  track  on  1|  mile } 

14.  How  much  is  4500  plus  -^-  of  4500  .? 

15.  Suppose  a  rail- way  car  to  run  350  miles  a  dav; 
what  distance  will  it  run  in  5 1  days  } 

16.  How  much  is  5-|  times  350.^ — That  is, —  how 
much  is  5  times  350  and  |  of  another  time  350  ^ 

17.  How  much  is  6|  times  91.^  S^  times  146^ 
3|  times  244  }     12|  times  379  }     lO-j^o  times  978  } 


PERCENTAGE. 

The  term,  per  cent.^  is  an  abbreviation  of  per  centum, 
and  signifies,  by  the  hundred.     I  per  cent,  of  any  num- 
ber, is  y-J-Q  of  that  number;  2  per  cent,  is  -jfo-;  3  per- 
cent, is  Yoo?  ^  P^'*  cent,  is  -j^^;  and  so  on. 
18.  What  is  5  per  cent,  of  360  dollars.^ 

3  Q  Q  Since  5  per  cent,  is  ^f^  j  we  multiply 

5  by  5,  and  divide  by  100.     To  divide 

■ by  100,  we  merely  point  off  two  figures 

$18.00         fj.Qj^  |.j^g  right,  as' taught  in  page  116. 


164  WRITTEN    ARITHMETIC.  VI. 

19.  What  is  1  per  cent,  of  100  ?     2  per  cent,  of  100  ? 

20.  What  is  2  per  cent,  or  i^q  of  350  dollars  ? 

21.  A  merchant,  who  has  $3875  deposited  in  the 
oank,  wishes  to  draw  out  4  per  cent,  of  his  deposit. 
How  manj  dollars  must  he  draw  ? 

22.  What  is  6  per  cent,  or  i|^  of  4250  dollars  ? 

23.  What  is  6  per  cent,  of  $92.50,  or  9250  cents  ? 

24.  What  is  4  per  cent,  of  $  132.75  ? 

25.  A  merchant  having  2513  gallons  of  wine  on  hand, 
lost  1  per  cent,  of  the  whole,  by  leekage  from  the  casks. 
How  many  gallons  did  he  lose  ? 

26.  Find  6  per  cent,  of  128  dollars. 

^28  After    multiplying  and    dividing,    our 

"  Q         quotient  is  $  7^^^^  .     Now,  since  j^q  of  a 
— —         dollar  is   1  cent,  -^^-^  of  a  dollar  is  63 
$7.68         cents:  therefore  the  answer  is  $7.68. 

27.  Find  7  per  cent,  of  2517  dollars. 

28.  Find  18  per  cent,  of  20  dollars. 

29.  What  is  4  per  cent,  of  $70.14,  or  7014  cents  ? 
In  this  example,  after  multiplying  by  4,  and  dividing 

by   100,  there  is  a  remainder  of  56,     And    since  the 
quotient  is  cents^  this  remainder  is  yoq  of  a  cent. 

30.  What  is  9  per  cent,  of  $470.46  ? 

31.  What  is  55  per  cent,  of  $964.07.? 

32.  A  and  B  have  $500  apiece.  If  A  should  give  B 
6  per  cent,  of  his  cash,  what  would  each  then  have  ? 

33.  What  is  ^  of  1  per  cent,  of  62  dollars  ? 

34.  What  is  4  7]  per  cent,  of  62  dollars  .? 

35.  What  is  ■}  per  cent,  of  246  dollars  } 

36.  What  is  5  J-  per  cent,  of  246  dollars  ?  • 

37.  A  merchant  paid  $491  for  a  quantity  of  salt:  for 
how  much  must  he  sell  it,  to  gain  9  per  cent..'^ 

38.  A  trader  paid  $230  for  a  piece  of  cloth,  contain- 
mg  46  yards,  and  sold  it  so  as  to  lose  4  per  cent. 
At  how  much  did  he  sell  it  per  yard  ? 

39.  If  I  pay  $525  for  90  barrels  of  flour,  at  what 
price  per  barrel  must  I  sell  it,  to  gain  7  per  cent..'' 

40.  Suppose  a  merchant  to  pay  $85  per  ton  for  6 
tons  of  iron;  at  what  price  must  he  sell  it  per  hundred- 
weight, in  order  to  gain  12  per  cent.  .'^     . 


23  INTEREST.  165 

41.  A  merchant  failed  in  business,  and  was  able  to 
pay  his  creditors  only  65  per  cent,  of  their  demands. 
What  did  he  pay  on  a  demand  of  ^  534  r 

INTEREST. 

Interest  is  money  paid  for  the  use  of  money  that  has 
been  owed.  For  instance,  suppose  that  A  lends  B  $  100 
for  one  year,  and  at  the  end  of  the  year,  B  pays,  not 
only  the  $  100,  but  also  pays  $  6  for  the  use  of  the  $  100: 
in  this  case,  $  6  is  the  interest. 

The  money  for  which  interest  is  paid,  is  called  the 
Principal  The  sum  per  cent,  paid  for  one  year's  in- 
terest, is  called  the  Rate,  The  principal  and  interest 
added  together,  are  called  the  •Amount. 

RULE  FOR  COMPUTING  INTEREST.  Mnltiply  the 
principal  by  the  rate  per  cent.^  and  divide  the  product 
by  100:  the  quotient  will  be  the  interest  for  1  year, 

42.  What  is  the  interest  of  $  100,  for  1  year,  at  5  per 
cent.  ?     What  is  the  amount  ? 

43.  What  is  the  interest  of  $  1  or  100  cents,  for  1 
year,  at  5  per  cent..'^     What  is  the  amount  ? 

44.  What  is  the  interest  of  $354,  at  6  per  cent,  for 
1  year  r  for  2  years  ?  for  3  years  ?  for  4  years  ? 
What  is  the  amount  for  4  years  ? 

45.  What  is  the  interest  of  |^ 40. 50,  for  4  years,  at  6 
per  cent.  ?     What  is  the  amount  ? 

46.  What  is  the  interest  of  $18,  for  3  years,  at  7 
per  cent.  ?     What  is  the  amount  ? 

47.  What  will  $8110  amount  to  in  15  years;  the  rate 
of  interest  being  4  per  cent.  ? 

48.  What  is  the  interest  of  $6470,  for  3  years,  at  5^ 
per  cent.?     What  is  the  amount  ? 

When  interest  is  to  be  computed  for  any  number  of  ^ 
months^ — First  find  the  interest  for  1  year;  then  take  -^^ 
of  a  yearns  interest  for  1  month;  j^  or  \  for  2  months; 
h  ^^  h  fi^  ^  months;  and  so  on, 

49.  What  is  the  interest  of  $35,  for  one  month,  at  6 
per  cent,  per  annum  ?     What  is  the  amount } 


106  WRITTEN    ARITHMETIC  Vl 

50.  What  is  the  interest  of  ^21,  for  2  months,  at  7 
per  cent,  per  annum  ? 

51.  What  is  the  interest  of  5^4291,  for  3  months,  at 
5  per  cent,  per  annum  ? 

52'  At  4  per  cent,  per  annum,  what  is  the  interest  of 
jjj  122.75  for  4  months  ?  for  5  months  ?  for  6  months  ? 
for  7  months  ?  for  8  months  ?  for  9  months  }  for  10 
months  ?     What  is  the  amount  for  11  months  ? 

53.  What  is  the  interest  of  $  14.50,  for  1  year  and  1 
month,  at  6  per  cent..^ 

54.  What  is  the  interest  of  $19.25,  for  3  years  and 
2  months,  at  8  per  cent.? 

55.  What  is  the  amount  of  $458,  for  2  years  and  3 
months,  at  7  per  cent..'* 

56.  What  is  the  amount  of  $8.75  for  5  years  and  4 
months,  at  4  per  cent.  ? 

57.  What  is  the  amount  of  $91.50,  for  2  years  and  7 
months,  at  8  per  cent..'^ 

58.  What  is  the  interest  of  $81,  from  February  7, 
1832,  to  August  7,  1835,  at  6  per  cent..^ 

59.  Suppose  a  promissory  note  of  $  145,  to  be  dated, 
January  15,  1 831 ;  what  will  be  the  amount  of  that  note, 
October  15,  1834;  the  rate  being  6  per  cent.."^ 

60.  A  owed  B  $96,  on  interest  at  6  per  cent.  At 
the  end  of  2  years,  A  paid  the  interest  then  due,  and  $25 
of  the  principal:  at  the  end  of  3  years  and  11  months, 
he  paid  the  whole  debt.     What  was  each  payment  ^ 

When  interest  is  to  be  computed  for  any  number  of 
days^ — First  find  the  interest  for  1  month;  then  take  -j^ 
of  a  month'' s  interest  for  1  day;  -f^  or  yj  for  2  days; 
io  or  i^for  3  days;  3%  or  -f^  for  4  days;  -^%  or  ^  for  5 
days;  -^^  or  \  for  6  days;  and  so  on. 

In  the  following  operations,  in  this  section,  all  fractions 
of  a  cent  may  be  disregarded:  this  being  the  common 
practice  in  business. 

61.  What  is  the  interest  of  $231,  for  7  days,  at  6 
per  cent,  per  annum  ? 

Direction.  First  find  the  interest  for  1  year;  then  tor 
i;^2  of  a  year  or  1  month;    and  then  for  -^q  of  a  month. 


I 


^3  INTTEREST.  167 

(12.  What  is  the  interest  of  $75,  for  10  days,  at  6 
per  cent,  per  annum  ? 

63.  What  is  the  interest  of  $254  for  21  days,  at  6 
per  cent,  per  annum  ? 

64.  What  is  the  interest  of  $  110,  for  5  months,  and 
.8  days,  at  6  per  cent,  per  annum  ? 

65.  What  is  the  interest  of  $34  for  1  year,  3  months, 
and  25  days,  at  6  per  cent,  per  annum  ? 

66.  What  is  the  interest  of  $91.18,  for  3  years,  2 
months,  and  13  days,  at  6  per  cent,  per  annum  } 

Several  other  methods  are  practised  by  merchants, 
in  computing  interest;  among  which,  are  the  following. 

When  the  rate  is  5  per  cent. — Divide  the  principal  by 
20,  and  the  quotient  is  the  interest  for  1  year. 

61.  What  is  the  interest  of  $4207,  for  2  years,  at  5 
per  cent,  per  annum  } 

63.   What  is  the  interest  of  $951.17,  for  4  years,  at 

5  per  cent,  per  annum  ? 

When  the  rate  is  6  per  cent. — Multiply  the  principal 
by  half  the  number    of  months  in  the  time.,  divide  the 
•product  by  100,  and  the  quotient  is  the  interest. 

69.  What  is  the  interest  of  $119,  for  16  months,  at 

6  per  cent., per  annum  ? 

70.  What  is  the  interest  of  $96.48,  for  10  months,  at 
6  per  cent,  per  annum  ? 

71.  What  is  the  amount  of  $27.56,  on  interest  6 
months,  at  6  per  cent,  per  annum  ? 

72.  What  is  the  interest  of  $133.24,  for  11  months, 
at  6  per  cent,  per  annum  ? 

To  find  the  interest  for  days,  the  rate  being  6  per 
cent.— Multiply  the  principal  in  dollars  by  the  number 
of  days.,  divide  the  product  by  6,  and  cut  off  one  figure 
from  the  right  of  the  quotient.  The  rest  of  the  quotient 
figures  express  nearly  the  interest^  in  cents. 

73.  What  is  the  interest  of  $249,  for  75  days,  at  6 
per  cent,  per  annum  ? 

74.  What  is  the  interest  of  $5824,  for  21  days,  at  6 
per  cent,  per  annum  ? 


168  WRITTEN    ARITHMETIC.  VI. 

75.  What  difference  will  it  make  to  the  man  who  pays 
interest  on  $  100  for  1  year,  whether  it  be  computed  by 
daysy  or  according  to  true  rule  in  page  165? 

DISCOUNT. 

Discount  is  an  ^ibatement  of  a  certain  part  of  a  debt, 
when  the  debt  is  paid  before  it  becomes  due.  For  in- 
stance ;  suppose  that  A  is  bound  to  pay  B  $  106,  in  one 
year  from  the  present  time ;.  but  B',  wanting  the  money 
now,  agrees  to  receive  $  100  for  the  debt,  on  condition 
of  present  payment:  in  this  case,  $100  is  the  present  worth 
of  the  debt,  and  $6,  is  the  discount. 

The  present  worth  of  any  debt  due  at  a  future  period, 
is  that  Slim  of  money,  which,  if  put  at  interest,  would 
amount  to  the  debt  by  the  time  it  becomes  due.  There- 
fore^  when  the  interest  is  5  per  cent.,  that  is,  j-^-^  of  the 
pnncipal,  then  the  discount  is  ^^^  of  the  principal. 

RULE  FOR  COMPUTING  DISCOUNT.  Multiply  the 
principal  by  the  number  of  cents  found  to  be  the  interest 
of  one  dollar  for  the  time,  and  divide  the  product  by  the 
number  which  results  from  adding  100  to  the  multiplier. 
The  quotient  will  be  the  discount. 

For  example,  at  the  rate  of  12425 

6  per  cent.,  the  discount  on  '  y\ 

$124.25,  due  in  1  year  and  — — ■ 

10  months,  is  found  thus—  HI)  1366.75(13.21i\\ 

76.  What  is  the  discount  on  $48.51,  due  in  3  years; 
the  rate  of  interest  being  5  per  cent.  ? 

77.  What  is  the  discount  on  $247,  due  in  1  year,  the 
rate  of  interest  being  6  per  cent.  ? 

78.  What  is  the  present  worth  of  $150,  due  in  1  year, 
the  rate  of  interest  being  6  per  cent.  ? 

Find  the  discount,  and  subtract  it  from  the  debt. 

79.  What  is  the  present  worth  of  $1640,  due  in  2 
years,  the  rate  of  interest  being  5  per  cent.  ? 

80.  Find  the  difference  between  the  discount  and  the 
interest  of  $  100  for  1  year,  the  rate  being  6  per  cent. 

81.  Find  the  present  worth  of  $75,  due  in  2  years  and 
9  months,  [2f  years],  interest  being  6  per  cent. 


24.  25.  26.  FRACTIONS  169 

Section  24. 

1.  Suppose  I  of  a  piece  of  broad-cloth  to  be  worth 
$118.87;  what  is  \  of  the  piece  worth  .^  What  is  the 
whole  piece  worth  ? 

2.  11887  is  I  of  what  number? 

3.  If  the  interest  of  $  100  be  $3.50  for  -f^  of  a  year, 
what  is  the  interest  of  $100  for  -^^  of  a  year.'*  Then 
what  would  be  the  interest  for  1  year  ? 

4.  If  -J-|  of  an  acre  of  land  produce  133  bushels  of 
potatoes,  how  many  bushels  does  t^-^  of  an  acre  produce  ^ 
Plow  many  bushels  would  1  acre  produce  ? 

5.  9071  is  ^Q  of  what  number.^ 

6.  If  a  man  earn  $190  a  yenr  by  working  y^^  of  the 
time,  how  much  could  he  earn  by  working  constantly  ^ 

7.  $  14  is  8  per  cent,  or  yf^-  of  what  sum  of  money  .'* 

Section  25. 

CHANGE  OF  THE  TERMS  OF  FRACTIONS. 
The  numerator  and  denominator  of  a  fraction,  are 
called  the  two  terms  of  a  fraction.  These  terms  maybe 
changed,  and  the  fraction  may  still  express  the  same 
quantity.  For  instance,  the  terms  2  and  3,  in  the  frac- 
tion f ,  may  be  changed  to  4  and  6,  and  the  fraction  will 
become  -J,  which  is  still  equal  to  f . 

1.  I  is  equal  to  how  many  twenty-fourths  ? 
Direction.     8-eighths  are  equal  to  24-twenty-fourths; 

therefore,  find  |-  of  24,  and  this  number  will  be  the  re- 
quired numerator  of  2-4- 

2.  Y  is  equal  to  how  many  fourteenths  ? 

3.  Change  f  to  eighteenths  and  add  y%  to  it. 

4.  J  is  equal  to  how  many  forty-fifths  ? 

5.  Change  ^^  to  fortieths,  and  then  take  -}^  from  it. 

Section   26. 

REDUCTION  OF  FRACTIONS  TO  LOWER    TERMS. 

When  a  number  can  be  found,  that  will  divide  both 
terms  of  a  fraction,  without  a  remainder,  the  two  quo- 
tients arising  from  the  division,  will  express  the  fraction 
redxiced  to  lower  terms.     For  example,  both  terms  of 

p 


170  WRITTEN  ARITHMETIC.  VI 

the  fraction  -^2  ^^"  ^®  divided  by  3,  and  the  reduced 
fraction  will  be  f .  Again,  both  terms  off  can  be  divid- 
ed by  2,  and  the  reduced  fraction  will  be  ^ .  Thus  any 
fraction  may  be  reduced  to  its  lowest  terms^  by  repeat- 
edly dividing  the  terms,  until  no  number  will  divide 
them  both  without  a  remainder. 

1.  Reduce  each  of  the  following  fractions  to  its  low- 

oct   tPrmc         3  6  4  9  10  6  18  12 

est  teims.    -g-.    9.    y^-     12-    T^;     is-    2Y-    24\ 

2.  Reduce  each  of  the  following  fractions  to  its  low- 

*»Gt   t^rmc        1'^  20  100  6.0  0  1_0_  4.5.        _30_ 

CbL   LC1111&.       5-Q-.      -g-Q  .      Xoo  .       9  00*      Too*       90*       120  0  • 

Only  once  dividing  the  terms  of  a  fraction,  will  reduce 
it  to  its  lowest  terms,  if  we  use  the  greatest  common  di- 
visor^ that  is,  the  greatest  number  that  will  divide  both 
terms  without  a  remainder. 

TO  FIJVD  THE  GREATEST  COMMON  DIVISOR  of  twO 
numbers^ — Divide  the  greater  number  by  the  smaller^ 
then  divide  the  divisor  by  the  remainder;  and  thus  con- 
tinue  dividing  the  last  divisor  by  the  last  remainder^  till 
nothing  remains.  The  divisor  used  last  of  all^  will  be 
the  greatest  common  divisor, 

3.  Find  the  greatest  common  divisor  of  91  and  117. 
91)117(1  This  operation  is  perform- 

91  ed  according  to  the  direction 

'26)9\(S  above,  and  13  is  found  to  be 

•J  Q  the  greatest  common  divisor; 

— ^       ^  or    the    greatest   number  by 

13)26^2  which  91  and  117  can  be  di- 

z^  vided  without  a  remainder. 

4.  Find  the  greatest  common  divisor  of  15  and  235. 

5.  Reduce  -^  to  its  lowest  terms,  by  using  the  great- 
est common  divisor  of  the  two  terms. 

6.  Reduce  to  their  lowest  terms,  |ff ,  ^|-|,  and  ||f 

Section  27. 

COMPOUND    FRACTIONS. 

A  compound  fraction  arises  from  dividing  a  unit  mto 
a  certain  number  of  equal  parts,  and  then  dividing  one 
of  these  parts  into  other  equal  parts. 


27.  FRACTIONS.  ^71 

TO  REDUCE  S  COMPOUJVD  FRACTION  TO  A  SIMPLE 
FRACTION^ — Multiply  all  the  numerators  together  for  a 
new  numerator^  and  all  the  denominators  for  a  new  denom- 
inator: then  reduce  the  new  fraction  to  its  lowest  terms. 

1.  Reduce  f  of  ^  to  a  simple  fraction. 

2.  ^  of  a  water  melon  was  divided  equally  among  6 
boys.     What  fraction  of  the  melon  did  1  boy  receive  .'* 

3.  Reduce  f  of  f  to  a  simple  fraction. 

4.  J  of  an  acre  of  land  was  divided  into  4  equal  lots. 
What  fraction  of  an  acre  did  2  lots  contain  ? 

5.  Reduce  |-  of  ^  to  a  simple  fraction. 

6.  I  of  1^4  is  equal  to  what  part  of  1  ? 

7.  Reduce  -^-j  of  i\  to  a  simple  fraction. 

8.  1  penny  is  what  part  of  Is.?     what  part  of  £1? 

9.  7  pence  is  what  simple  fraction  of  J£l  .'^ 
Suggestion.     7  pence  is  -12  of  1  shilling,  and  1  shilling 

IS  ^  of  £1.     Therefore,  7  pence  is  j^  ^^  2V  of  ^1» 

10.  Reduce  10  grains  to  the  fraction  of  an  ounce; 
that  is,  reduce  ^^  of  -^q  to  a  simple  fraction. 

11.  Reduce  3  nails  to  the  fraction  of  ^  yard. 

12.  Reduce  4  inches  to  the  fraction  of  a  yard. 

13.  Reduce  25  seconds  to  the  fraction  of  an  hour. 

14.  Reduce  f  of  :j  of  |^  to  a  simple  fraction. 

15.  4  of  f  of -j^Q  is  equal  to  what  part  of  1  .'* 

16.  Reduce  -^^  of  ^f  ^^  ts  ^^  ^  simple  fraction. 
When  the  lower  denominations  of  a  compound  number 

are  to  be  reduced  to  the  fraction  of  a  higher  denomina- 
tion^—  First ^  reduce  the  given  quantity  to  the  lowest  de- 
nomination mentioned^  and  this  number  will  be  the 
numerator:  then  reduce  a  unit  of  the  higher  denomina- 
tion^ to  the  same  denomination  with  the  numerator^  and 
this  number  will  be  the  denominator. 

17.  Reduce  14s.  lOd.  2qr.  to  the  fraction  of  ,£1. 
14s.  lOd.  2qr.      £\  is  20s.  The  denomin- 
12                                     12          ^^or  and  numer- 

T^A  77777^  ator  here. found, 

^'f  ^''f  make  IB:    this 

. fraction,whenre- 

Awm.714qr.  Denom.  960  qr.  duced,  is  -^. 


172  WRITTEN    ARITHMETIC.  VI 

18.  Reduce  2s.  7d.  Iqr.  to  the  fraction  of  c£l. 

19.  Reduce  lid.  3qr.  to  the  fraction  of  a  pound. 

20.  Reduce  15s.  Od.  3qr.  to  the  fraction  of  a  pound 

21.  Reduce  lOd.  Iqr.  to  the  fraction  of  a  shiUing. 

22.  Reduce  2s.  9d.  3|^qr.  to  the  fraction  of  a  pound. 
Direction,     Find  the  number  of  fifths  of  a  farthing  in 

8s.  9d.  3f  qr.,  for  a  numerator;  then  find  the  number  of 
fifths  of  a  farthing  in  £1,  for  a  denominator. 

23.  Reduce  Sj  pence  to  the  fraction  of  a  pound. 

24.  Reduce  5qt.  Ipt.  to  the  fraction  of  a  bushel. 

25.  Reduce  9gal.  3qt.  Ipt.  to  the  fraction  of  Ihhd* 

26.  Reduce  6  rods  3yd.  2ft.  to  the  fraction  of  a  mile. 

27.  Reduce  35 -jq  seconds  to  the  fraction  of  a  day. 

When  the  fraction  of  a  higher  denomination  is  to  be 
reduced  to  its  value  in  whole  numbers  of  lower  denomi- 
nation^—  Multiply  the  numerator  by  that  number  of  the 
next  lower  denomination  which  is  required  to  make  a  unit 
of  the  higher^  and  divide  the  product  by  the  denominator; 
the  quotient  will  be  a  whole  number  of  the  lower  denomi- 
nation^ and  the  remainder  will  be  the  numerator  of  a  frac- 
tion.    Proceed  with  this  fraction  as  before^  and  so  on, 

28.  Reduce  f  of  £1  to  its  value  in  shillings  &c. 

2  Since  f  of  £  1  is  the  same  as  f  of 

20  20  shillings,  we  find  f  of  20  shillings, 

»7\^Q  in  shillings  and  the  fraction  of  a  shil- 

—  ling; —  itis  5|shillings.     Then,  since 

^    ^  •  I  of  1  shilling  is  the  same  as  ^  of  12 

^^  pence,  we  find  f  of  12  pence; —  it  is 

7)60  8f  pence.     Then,  since  f  of  1  pen- 

"~^   4  ny  is  the  same  as  |  of  4  farthings, 

4  we  find  ^  of  4  farthings; — it  is  2f 

—         farthings.     Thus  by  finding  one  de- 

'^/l^^       nomination  at  a  time,  we  finally  ob- 

2f        tain,  5s.  8d.  2fqr. 

29.  Reduce  f  of  ^  1  to  its  value  in  shillings  &c. 

30.  I  of  £1  is  how  many  shillings,  pence,  &c..'* 

31.  In  I  of  a  shilling,  how  many  pence,  &c..^ 

32.  Change  £15  f  to  pounds,  shiUings,  pence,  Slc 
S3.  Reduce  ^  of  Icwt.  to  quarters,  pounds,  &c. 


28.  FRACTIONS.  173 

34.  Change  9^^  pounds,  to  pounds,  ounces,  and  drams. 

35.  Reduce  |  of  a  mile  to  furlongs,  rods,  feet,  &c. 

36.  In  lOf  acres,  how  many  acres,  roods,  rods,  &c. 

37.  How  many  dimes,  cents,  and  mills,  in-/2  of  $1  .^ 

38.  In  -j\  of  a  dollar,  how  many  cents  and  mills  ? 

39.  Suppose  sugar  to  be  $  12  per  hundredweight; 
what  quantity  can  be  purchased  for  $  113  .^ 

Section  28. 

\ 

COMMON  DENOMINATORS. 
When  two  or  more  fractions  have  the  same  number 
for  a  denominator,  this  number  is  called  their  Common 
Denominator,  Fractions  having  different  denominators, 
must  be  reduced  to  a  common  denominator,  before  ad- 
dition or  subtraction  can  be  performed  on  them. 

RULE  FOR  REDUCING  FRACTIONS  TO  A  COMMON 
DENOMINATOR.  Miiliiply  each  numerator  into  all  the 
denominators  except  its  own^  for  a  new  numerator.  Then 
multiply  all  the  denominators  together  for  a  common  de- 
nominator^  and  place  it  under  each  new  numerator. 

1.  Reduce  |,  f  5  and  f,  to  a  common  denominator. 

5  4  6  8 

.         _9  ^         __8_  9 

45  32  48  72 

_7  _I         _^  Z. 

315  224  432  ^04- 

2.  Reduce  f ,  1% ,  and  \  to  a  common  denominator. 

3.  Reduce  \\  and  -^-^  to  a  common  denominator. 

4.  Reduce  ^,  :! ,  and  -3-^  to  a  common  denominator. 

5.  Reduce  -^,  -j,  f,  and  \  to  a  common  denominator. 

6.  How  much  is  f  and  y  added  together  ? 

7.  How  much  is  ^-f  and  i|  added  together  ? 

8.  How  much  is  f  and  |  and  -{^  added  together  }     . 

9.  If  ^5  be  taken  from  |,  how  much  will  remain  ? 

10.  If  f  be  taken  from  |f ,  how  much  will  remain  } 

1 1 .  Which  is  greater,  -/g  or  i\  ? — how  much  greater  ? 


174  WRITTEN    ARITHMETIC.  VI. 

Section  29. 

1.  A  farmer  raised  142 1  bushels  of  corn  m  one  field, 
and  237y^2  bushels  in  another.  How  many  bushels  did 
he  raise  in  both  fields  ? 

The  learner  may  reduce  the  fraction  in  his  answer,  to 
Its  lowest  terms,  in  this,  and  all  future  examples. 

2.  A  farm  is  divided  into  three  lots; — the  first  lot  con- 
taining 46 1-  acres,  the  second  50 f  acres,  and  the  third 
62yq  acres.     How  many  acres  are  there  in  the  farm.'' 

3.  Add  together  441|  and  65^  and  2556^. 

4.  If  6f  be  taken  from  8  f  how  much  will  remain  } 

5.  Subtract  437|  from  1659 f. 

6.  If  6  I  gallons  of  wine  should  leak  from  a  cask  con- 
taining 53  "I  gallons,  how  many  gallons  would  remain  ? 

7.  Add  together  623 1  and  113j^2  5  ^^^^  ^^en  subtract 
from  the  sum  450  f. 

8.  Three  soldiers  shared  a  loaf  of  bread  as  follows: — 
the  first  took  f  of  it,  the  second  took  yj  of  it,  and  the 
third  took  the  remainder.  What  fractional  part  of  the 
loaf  did  the  third  soldier  receive  ? 

9.  A  trader  having  25  barrels  of  flour,  sold  8|-  barrels 
to  one  man,  and  9||  barrels  to  another.  What  quantity 
of  flour  had  he  then  remaining  ? 

Section  30. 

1.  Suppose  I  have  16  dollars;  to  how  many  men  can 
I  give  f  of  a  dollar  apiece  ? 

2.  How  many  times  is  f  contained  in  16  ? 

3.  How  many  pairs  of  gloves  can  I  buy  for  18  dollars, 
the'  price  being  |  of  a  dollar  a  pair? 

4.  Divide  18  by  | ;  that  is,  reduce  18  to  fourths^  and 
find  how  many  times  3-fourths  is  contained  therein. 

5.  Divide  46  by  f ;  that  is,  find  how  many  times  the 
fraction  f  is  contained  in  46. 

6.  If  a  man  walk  1  mile  in  ^%  of  an  hour,  what  dis- 
tance will  he  walk  in  4  hours  ? 

7.  How  many  times  is  jq  contained  in  4? 

8.  How  many  times  is  f  contained  in  ^  ? 
Direction.     Reduce  f  and  -^  to  a  common  denomina- 
tor, and  then  divide  one  n  imerator  by  the  other. 


29.  30.  31.  FRACTIONS.  175 

9.  How  many  times  is  -^j  contained  in  -^  ? 

10.  How  many  barrels  of  flour  can  be  bought  for  38 
dollars,  at  4|  dollars  per  barrel  ? 

1 1 .  How  many  times  is  25  f  contained  in  91  f  .'^ 

12.  How  many  times  is  6|  contained  in  423-? 
Direction.     Reduce  the  two  fractions  to  a  common 

denominator;  then  reduce  the  mixed  numbers  to  impro- 
per fractions,  and  divide  one  numerator  by  the  other. 

13.  If  a  barrel  of  cider  will  last  a  man  3|  monlhsj 
how  manv  barrels  will  he  drink  in  lOf  months  ? 

Section  31. 

REVIEW. 

1.  ( §  13.)  Suppose  a  man  to  earn  95  cents  per  day; 
how  many  days  would  it  take  him  to  earn  $43.16  ? 

2.  ( §  14.)  How  many  pounds  of  sugar  can  be  bought 
for  $  14|,  when  the  price  is  |^  of  a  dollar  a  pound  ? 

3.  (§15.)  If  1  yard  of  silk  cord  cost  -^^  of  a  dollar, 
What  is  the  price  of  75  yards,  at  the  same  rate  ? 

4.  (§16.)  Ifl4|-  yards  of  cloth  be  cut  from  a  piece 
containing  52 1-  yards,  how  much  will  be  left  ? 

^*  (§  ^'^O  ^f  ^  pound  of  shot  cost  -^q  of  a  dollar, 
what  will  be  the  cost  of  17  pounds,  at  the  same  rate  ? 

6.  (§18.)  How  much  corn  will  grow  on  140  acres 
of  land;  allowing  each  acre  to  produce  34 1  bushels  ? 

7.  (§  19.)  If  a  man's  expenses  be  $46.24yq  for -^ 
of  a  year,  what  will  be  his  expenses  for  1  year  ? 

8.  (§20.)  ^  of  71  is  equal  to  what  improper  frac- 
tion ?     Then  ^  of  71  is  equal  to  what  mixed  number  ? 

9.  (§21.)  Suppose  42  men  to  share  equally  in  a 
prize  of  $  1000;  what  is  the  share  of  one  man  .'' 

10.  ( §  22.)  If  a  man  can  cut  46  cords  of  wood  in  14 
days,  how  many  cords  can  he  cut  in  60  days  ? 

11.  (§23.)  Suppose  a  man  can  build  a  mile  of  wall 
in  310  days;  in  what  time  can  he  build  -^  of  a  mile  ? 

12.  (§24.)  A  man  bought  a  quantity  of  flour,  for 
domestic  use,  and  in  36  days  he  found  that  ^  of  it  was 
consumed.     How  long  would  the  whole  last  ? 


176  WRITTEN    ARITHMETIC.  VI 

13.  (§25.)  f  is  equal  to  how  many  ninths.?  how 
many  thirty-sixths  ?      how  many  seventy-fifths  ? 

14.  (§  26.)  Reduce  the  two  fractions,  ^  and  ff,  to 
their  lowest  terms,  and  then  add  them  together. 

15.  (§27.)  Suppose  a  farm  to  contain  98^^^  acres  of 
land;  how  many  acres  are  there  in  |  of  the  farm  ? 

16.  (§  28.)  Add  together,  -^^  and  -j^  and  \i',  then 
subtract  j^  froni  the  sum; — what  is  the  remainder  ? 

17.  (§29.)  If  f  and  -f^  of  a  number  be  subtracted 
from  itself,  what  part  of  that  number  is  the  remainder  ? 

18.  (§  30.)  If  the  mail-stage  run  9^^  "^iles  in  1  hour, 
how  many  hours  will  it  be  in  running  175  f  miles  ? 

RETROSPECTIVE   OBSERVATIOjYS. 

A  fraction  is  rendered  greater  by  increasing  the  nu- 
merator, and  smaller  by  increasing  the  denominator. 

To  multiply  a  fraction  by  a  whole  number^ —  Either 
multiply  the  numerator^  or  divide  the  denominator. 

To  divide  a  fraction  by  a  whole  number^ — Either  di' 
vide  the  numerator^  or  multiply  the  denominator. 

When  a  number  is  multiplied  by  1,  the  product  is 
equal  to  the  multiplicand.  Therefore,  when  a  number 
is  multipHed  by  a  fraction,  which  is  less  than  1,  the  pro* 
duct  must  be  less  than  the  multiplicand. 

To  multiply  a  whole  number  by  a  fraction^ — Multiply 
by  the  numerator^  and  divide  by  the  denominator. 

Dividing  a  number  by  1,  gives  a  quotient  equal  to  the 
dividend.  Therefore,  dividing  a  number  by  a  proper 
fraction,  must  give  a  quotient  greater  than  the  dividend, 
because,  the  fraction  being  less  than  1,  is  contained  a 
greater  number  of  times  in  the  dividend. 

To  divide  a  whole  number  by  a  fraction^ — Multiply 
by  the  denominator,,  and  divide  by  the  numerator. 

To  multiply  a  fraction  by  a  fraction^ — Multiply  nu- 
merator by  numerator,,  and  denominator  by  denominator. 

To  divide  a  fraction  by  a  fraction,, — Multiply  the  nu- 
merator of  the  dividend  by  the  denominator  of  the  divisor^ 
for  a  numerator;  and  multiply  the  denominator  of  the  div' 
idcnd  by  the  numerator  of  the  divisor^  for  a  denominator 


32  MISCELLANEOUS    EXAMPLES.  177 

19.  Multiply  the  fraction,  yV^?  t>y  3S- 

20.  Divide  the  fraction,  y^,  by  145. 

21.  Multiply  S706  by  the  fraction,  |f|. 

22.  Divide  611  by  the  fraction,  -^^Iq  . 

23.  Multiply  the  fraction,  -^ff ,  by  the  fraction,  -^. 

24.  Divide  the  fraction,  2^,  by  the  fraction,  ^f . 

25.  What  is  the  product  of  608 f  multiphed  by  8-^.^ 
26  What  is  the  quotient  of  45  f  divided  by  3|? 

Section  32. 

MISCELLANEOUS    EXAMPLES. 

1.  Suppose  a  man  can  perform  a  journey  in  14  days 
and  3  hours,  travelling  9  hours  a  day;  in  what  time  can 
he  perform  the  journey,  travelling  11  hours  a  day? 

2.  A  trader  gave  5^75  for  56  gallons  of  wine,  and  lost 
11  gallons  by  leakage.  At  how  much  per  gallon  must 
he  sell  the  remainder,  to  get  the  whole  cost? 

3.  Suppose  a  retailer  to  pay  $165  for  a  ton  of  sugar, 
at  what  price  must  he  sell  it  per  pound,  in  order  to  gain 
10  per  cent,  on  the  cost? 

4.  What  quantity  of  salt,  worth  62  cents  per  bushel, 
must  be  given  in  exchange  for  258  pounds  of  pork, 
worth  9  cents  per  pound? 

5.  What  is  the  profit  on  400  hogsheads  of  molasses, 
purchased  in  New  Orleans  at  12-^  cents  per  gallon,  [63 
gal.  in  each  hhd.],  freighted  to  New  York  at  5^3.50  per 
hhd.,  and  sold  at  24  cents  per  gallon;  3gal.  2qt.  having 
leaked  from  each  hhd.  on  the  passage? 

6.  If  a  pint  of  rum  a  day  will  kill  a  man  in  a  year  and 
a  half,  how  many  men  would^  a  cargo  of  600  hogsheads 
kill  in  the  same  time? 

7.  If  1 1  young  men  can  become  fools  by  drinking  6 
bottles  of  wine,  at  $  3  a  bottle,  what  would  it  cost  a 
dinner  party  of  25,  to  become  fools  in  like  manner? 

8.  If  a  man's  expenses  be  $1.40  a  day,  and  his  in- 
come $700  a  year,  what  will  he  lay  up  in  7  years? 

9  A  and  B  are  laborers — A  earns  $19.50  a  month, 
and  B  earns  $  16.25;  but  A  gives  B  -y\  of  his  earnings 
What  will  each  lay  up  in  14  months? 

10.  Find  the  difference  between  |  of  91,  and  |- of  91 


32,  MISCELLANEOUS    EXAMPLES.  179 

On  the  opposite  page,  30  cities  and  towns  are  exhib- 
ited in  their  respective  situations,  relative  to  each  other; 
and  the  number  of  miles,  by  mail-road  from  town  to 
town,  is  noted  in  figures. 

11.  Find  the  distance  from  Washington,  through  the 
mtermediate  towns,  to  Augusta,  Me from  Washing- 
ton to  Detroit from  Washington  to  St.  Louis 

from  Washington  to  Natchez from  Washington  to 

New  Orleans from  New  Orleans  to  Augusta,  Me. 

12.  Suppose  a  citizen  in  each  of  the  places  on  the 
opposite  page,  to  start  for  Washington,  and  travel  7 
miles  an  hour,  10  hours  in  each  day;  how  long  will  each 
one  be  in  performing  his  journey  ^ 

13.  How  long  would  it  take  you  to  walk  from  your 
school-room  to  Washington;  allowing  that  you  could 
walk  3^  miles  an  hour,  7  hours  in  each  day  .'^ 

14.  Two  men  started  at  the  same  time — one  of  them 
from  New  Orleans,  and  the  other  from  Augusta,  Me. — 
and  travelled  towards  each  other,  with  equal  speed. 
Between  what  two  towns,  and  what  distance  from  each 
of  these  towns  did  they  meet  ? 

15.  Mr.  A.  went  from  Portland  to  Baltimore,  travel- 
ling 5  miles  an  hour,  and  10  hours  a  day,  Mr.  B.  per- 
formed the  same  journey;  but  started  1  day  later,  and 
tiuvelled  7  miles  an  hour.     Where  did  B.  pass  A.? 

16.  Divide  $1000  among  A,  B,  and  C,  giving  B 
twice  as  much  as  A,  and  C  twice  as  much  as  B. 

17.  Gunpowder  is  composed  of  5  parts  sulphur,  7 
parts  charcoal,  and  38  parts  nitre.  How  many  pounds 
of  each  ingredient,  in  100  pounds  of  powder  ? 

IS.  A  and  B  purchased  a  cow  for  $16.  A  paid  $9 
of  the  price,  and  B  paid  $7.  They  sold  the  cow  for 
$21.     What  was  each  one's  share  of  the  gain  ? 

Solution,  Since  A  paid  -ij  of  the  price,  and  B  -^. 
A  must  have  -/^-  of  the  gain,  and  B  -f^, 

19.  C  and  D  traded  in  partnership;  C  owned  $450 
of  the  stock  in  trade,  and  D  $290.  They  gained  $  146 
What  was  each  one's  share  of  the  gain  ? 

20.  Suppose  $1000  stock  in  trade  to  gain  $230: 
what  is  the  gain  on  $351  of  that  stock  .'^ 


180  WRITTEN    ARITHMETIC.  VI 

21.  E  and  F  purchased  245  acres  of  land,  for  $2600. 
E  paid  $1200  of  the  money,  and  F  paid  the  remainder. 
How  much  land  should  each  one  have? 

22.  The  national  debt  of  England  is  not  less  than 
$  1  900000  000.  Allowing  5  percent,  interest  to  be  paid 
on  this  sum,  how^  many  families  would  it  support,  each 
family  spending  5J5400  per  annum? 

23.  If  a  man  can  dig  a  trench  in  15  days,  and  a  boy 
can  dig  the  same  trench  in  18  days,  in  what  time  can 
they  both  dig  it?     (See  example  20,  Oral  sec.) 

24.  How  many  days  will  it  take  17  men  to  perform  a 
{>iece  of  work,  that  1  man  can  perform  in  95  days? 

25.  How  many  days  will  it  take  30  men  to  perform  a 
piece  of  work,  that  4  men  can  perform  in  50  days? 

26.  How  many  days  will  it  take  25  men  to  perform  a 
piece  of  work,  that  6  men  can  perform  in  40  days? 

27.  If  15'  yards  of  carpeting,  which  is  one  yard  wide, 
will  cover  the  floor  of  a  room,  how  many  yards  3f  car- 
peting, 3-quarters  wide  will  cover  the  same  floor? 

Direction.  Find  the  number  of  square  quarters  con- 
tained in  15  yards  of  the  wider  carpeting;  then  divide 
this  number,  by  the  number  of  square  quarters  contained 
in  one  yard  of  the  narrower  carpeting. 

28.  Suppose  3^  yards  of  broad-cloth  5-quarter9 
wide,  to  be  made  into  a  cloak;  how  many  yards  of  silk 
3-quarters  wide,  will  it  take  to  line  the  cloak? 

29.  How  many  yards  of  carpeting  that  is  5-quarters 
wide,  will  cover  the  floor  of  a  room  which  is  19  i  feet 
in  length,  and  15  feet  in  width? 

30.  How  many  bricks  will  it  take  to  build  a  vrall,  1 
foot  thick,  5  feet  high,  and  24  feet  long;  each  brick  be* 
mg  8  inches  long,  4  inches  wide,  and  2  inches  thick? 

31.  If  a  man  can  hoe  fg  ^f  an  acre  of  corn  in  a  day, 
and  a  boy  ^  of  an  acre,  how  much  can  they  both  hoe  in 
a  day?     In  what  time  can  they  both  hoe  9  acres? 

32.  There  is  a  cistern,  having  3  pipes;  the  first  pipe 
will  discharge  the  cistern  in  4  holirs,  the  second  in  5 
hours,  the  third  in  6  hours.  What  part  of  the  contents 
of  the  cistern  would  all  the  pipes  together  let  ofl^in  1  hour. 
In  what  time  would  they  all  discharge  the  cistern^ 


32.  MISCELLANEOUS    EXAMPLE?.  IQl 

33.  What  is  the  height  of  a  steeple,  whose  shadow  is 
148  feet  4  inches,  when  a  shadow  5  feet  4  inches  long  is 
projected  from  a  post  6  feet  4  inches  high  ? 

34.  A  trader  failed  in  business,  owing  $11000,  and 
having  only  $  5000  to  divide  among  his  creditors.  How 
much  did  he  pay  on  a  debt  of  $  95.20  ? 

35.  A  fox  has  50  rods  the  start  of  a  greyhound,  but  the 
hound  runs  15  rods  while  the  fox  runs  9-^.  How  many 
rods  must  the  hound  run,  to  catch  the  fox? 

36.  A  cubic  foot  of  air  weighs  1^  ounce.  How  many 
pounds  of  air  does  a  iXDom  contain,  which  is  16  feet  long, 
14  feet  wide,  and  10  feet  high? 

37.  What  number  must  that  be,  which,  being  increased 
by  its  half,  and  its  third,  becomes  88  ? 

38.  A  and  B  hired  a  pasture  for  §  30.  A  turned  in  3 
cows,  and  B  turned  in  12  sheep.  Allowing  5  sheep  to  be 
equal  to  1  cow,  what  must  each  pay  ? 

39.  Suppose  London  has  1  500  000  inhabitants.  New 
York  350  000,  Philadelphia  220  000,  New  Orleans 
115  000,  Baltimore  110  000,  and  Boston  105  000;  how 
many  times  greater  is  London,  than  each  of  the  others? 


When  B.  scholar  has  reached  this  point,  it  will  be  well  to  con- 
sider how  much  more  time  he  is  likely  to  devote  to  study.  If  he 
have  but  a  few  months  more  to  spend  in  school,  the  Supplement 
will  furnish  for  him  tlie  suitable  exercises,  with  which  to  finish  his 
course  of  study  in  arithmetic.  If,  however,  he  is  likely  to  continue 
at  school  for  several  years,  he  may  omit  the  Supplement,  and  enter 
immediately  upon  the  exercises  of  Part  Third. 

In  the  preceding  chapters,  departments  of  business  are  not  ar- 
ranged under  distinct  heads.  Tlie  arrangement  is  strictly  anth- 
Tnetical,  and  business  examples  are  made  incidental  to  tlie  course. 
£n  the  Supplement,  departments  of  business  are  separately  pre- 
sented, in  distinct  articles.  These  articles,  although  brief,  arc 
rendered  sufficient,  by  the  learner's  previous  familiarity  with  the 
operations  they  require. 


182 

SUPPLEMENT. 


Article  I.     f 

INDICATIVE    CHARACTERS    OR    SIGNS. 

-}-  (Plus,)  standing  between  numbers,  indicates  that 
they  are  to  be  added  together;  thus,  3-\-2  is  5. 

—  {Minus,)  indicates  that  the  number  after  it  is  to  be 
subtracted  from  the  number  before  it ;  thus,  5 — 2  is  3. 

X  (Into,)  indicates  that  one  number  is  to  be  muUiplied 
into  another  ;  thus,  4X3  is  12. 

~  {By,)  indicates  that  the  number  on  the  left  is  to  be 
divided  by  the  number  on  the  right;  thus,  12-7-3  is  4. 

=  (Equal  to,)  indicates  that  the  number  before  it  is 
equal  to  the  number  after  it ;  for  example,  4-[-2=i6. 
6~2z=i4.     5X3=:15.     15~3=i5. 

CANCELLATION    OF    FACTORS. 

The  cancellation  of  factors  is  the  excluding  of 
such  factors  from  an  operation  as  balance  each  other. 

Any  two  equal  factors,  one  being  a  Hictor  of  a  dividend, 
and  the  other  a  factor  of  the  divisor,  or,  one  a  factor  of  a 
numerator,  and  the  other  of  the  denominator,  may  be  can- 
cclled,  that  is,  crossed  and  omitted.  For  example,  ^  of  | 
of  i  is  reduced  to  a  simple  fraction,  as  follows  — 

Here  we  cancel  the  two  threes,       _L_z?_2_ i 

and  multiply  1  by  1,  and  4  by  2.       X    4      2    """  ®' 

When  one  of  two  opposite  factors  will  divide  the  other 
without  a  remainder,  both  may  be  cancelled,  and  the  quo- 
tient retained  in  the  place  of  the  factor  divided.  For  in- 
stance, let  us  find  what  is  |  of  |-  of  {J-  ^^^i  ^^  ^^^ 


2 

1         90 


XX  ^     1      20 

^X  11  X    1  ■"  ^"^  ""  ""^^ 

3 

1.  Reduce  |-  of  -|  of  \^  of  §  to  a  simple  fraction. 

2.  What  is  I  of  f  of  I  of  1-1  of  -X.  of  \  of  100  ? 

3.  Reduce  f  of  |  of  |  of  -{^  to  a  simple  fraction. 
When  all  of  a  term  is  cancelled  off,  the  new  term  must  be  1 


11,;  DECIMALS.  183 

4.  A  merchant  owning  ^  of  |-  of  f  of  a  ship^  sold  f  of  his 
share.     What  part  of  the  ship  did  he  s^ll  ? 

5.  3  men  owned  equally  a  saw-mill ;  one  sold  |  of  J  of 
I  of  his  share.     What  part  of  the  mill  did  he  sell  ? 


II. 

DECIMAL  FRACTIONS. 

A  DECIMAL  FRACTION  IS  a  fraction  whose  denominator  is 
10,  or  100,  or  1000,  &c.  The  denominator  of  a  decimal 
fraction  is  never  written :  the  numerator  is  written  with  a 
point  prefixed  to  it,  and  the  denominator  is  understood  to 
be  a  1,  with  as  many  ciphers  annexed  as  there  are  figures 
in  the  numerator.      Thus,   .3  is  -j^^  ;  .31  is  -^-^^'y  .316  is 

1.  Write  upon  the  slate,  the  decimals  expressing  the 

fnl  1  nwi n o-  fi'flPt inn <2  3         46         708         1642      _9_6J)_4JL 

loiiowmg  nactions.     -j^.  ^^q.  -jooo*  T(5ooo-  Tooooo- 

When  a  whole  number  and  a  decimal  are  written  to- 
gether, the  decimal  point  is  placed  between  them.  Thus, 
24.6  is  24j%;  5.71  is  Sf^V;  48.364  is  48^3_6_4_. 

2.  Write  the  following  mixed  numbers,  expressing  the 
fractions  decimally.     SS-j^^.  516^%%.  S^V^c.  24^Vo¥o- 

In  whole  numbers,  any  figure,  wherever  it  may  stand, 
expresses  a  quantity  ^^  as  great  as  it  would  express,  if  it 
were  written  one  place  further  to  the  left.  For  instance, 
in  the  number  1111,  the  1  hundred  is  -^^  of  a  thousand  ; 
the  1  ten  is  -j^^  of  a  hundred,  or  j^q  of  a  thousand ;  the 
1  unit  is  ^Q  of  a  ten,  or  j-^^^-^  of  a  thousand.  In  decimals, 
this  system  is  continued  below  the  place  of  units. 

For  example,  in  the  number  1.111,  the  ^-S 

1  next  to  the  right  of  the  unit  is  l-tenth,  l| 

that  is,  j^^  of  a  unit ;  the  1  next  to  the  right         .ii  f  c  § 
of  the  1-tenth  is  -^^  of  a  tenth,  or  l-hun-  oS^S 

dredth  of  a  unit;  the  one  next  to  the  right  §  §  g  § 

of  the  1-hundredth,  is  -J^  of  a  hundredth,  1111 

or  I'thousandih  of  a  unit. 

Ciphers  placed  on  the  right  hand  of  decimal  figures,  do 
not  alter  the  value  of  the  decimal ;  because,  the  figures 


184 


SUPPLEMENT. 


II. 


remain  unchanged  in  their  distance  from  the  unites  place. 
For  instance,  .5,  .50,  and  .500  are  of  equal  value ;  being 
each  equal  to  ^.  But  every  cipher  placed  on  the  left  of  a 
decimal,  renders  it  ten  times  smaller,  by  removing  the  fig- 
ures one  place  further  from  the  unit's  place.  Thus,  if  we 
prefix  one  cipher  to  .5,  it  becomes  .05  [j^q]  ;  if  we  pre- 
fix two  ciphers,  it  becomes  .005  [to^oI  '  ^"^  ^^  ^"' 

3.  Write  upon  the  slate,  decimals  expressing  the  fol- 
lowing fractions.  ^^.  ^^^^^,  ^f ^^.  ^^^%^^.  io otooo- 

To  READ  DECIMAL  FRACTIONS, — Enumer&te  and  read 
the  figures,  as  if  they  were  whole  numbers,  and  conclude 
by  pronouncing  the  name  of  the  lowest  denomination, 

4.  Copy  upon  the  slate,  and  read  the  following  decimals. 
.06  .065  .0007  24.02 

.008  .409  .06264  5.763084 

.013  .207862         .10809  160.052 

.0514         .5004  .6500171         712.3005 

5.  Write  in  decimals  the  following  mixed  numbers. 


Q    14 


7   3  oi_ 
'Toooo 


^OO^oW 


^Toooooo 

Ofi 15 

'^^10000000 


ADDITION    OF    DECIMALS. 

6.  Add  the  following  numbers  into  one  sum.     63.75 
and  524.0764  and  .23  and  261.803. 


63.75 
524.0764 

.23 
26L803 

849.8594 


In  arranging  decimals  for  addition, 
we  place  tenths  under  tenths,  hun- 
dredths under  hundredths,  &c.  We 
then  begin  with  the  lowest  denomina- 
tion, and  proceed  to  add  the  columns 
as  in  whole  numbers. 

7.  What  is  the  sum  of  2.164,   870.31,   756,   9.18, 
157.0008,  26.104,  and  .3728? 

8.  What  is  the  sum  of  2706,  58.2,  .2065,  6.441,  75, 
14.2,  and  990.752? 

In  Federal  Money,  the  dollar  is  the  unit ;  that  is,  dol- 
lars are  whole  numbers ;  dimes  are  tenths,  cents  are  hun 
dredths,  and  mills  are  thousandths.     See  page  124. 


II.  DECIMALS.  Ig5 

9.  Add  together  $  24.6,  $  9.07,  $  5.009,  and  5  cents. 

10.  Write  the  following  sums  of  money  in  the  form 
of  decimals,  and  add  them  together.  $  46  and  9  cents, 
14  cents,  $7  and  8  mills,  6  dimes,  8  dimes  and  7  mills. 

SUBTRACTION    OF   DECIMALS. 

11.  Subtract  52.6087  from  406.91. 

4Q591  After  placing  tenths  under  tenths, 

52.6087  ^^'J  ^^^  subtract  as  in  whole  numbers. 
'  J.  .  r.  ^ ,  o  The  blank  places  over  the  7  and  8, 
-  *^   *' — ^         are  viewed  as  ciphers. 

12.  Subtract  943.076  from  8270.54. 

13.  Subtract  1084.72  from  5603.0626. 

14.  Subtract  146.1706  from  16094. 

15.  Find  the  difference  between  .8  and  .08,  by  sub- 
tracting the  smaller  decimal  from  the  greater. 

16.  Find  the  difference  between  .45  and  .31067. 

17.  What  is  the  difference  between  1  and  .046? 

18.  Write  4  dollars  and  8  mills  in  decimal  form,  and 
subtract  therefrom,  6  dimes  and  5  mills. 

19.  Subtract  7  cents  and  3  mills  from  10  dollars. 

MULTIPLICATION    OF    DECI?JALS. 

Multiplying  by  any  fraction,  is  taking  a  certain  part 
of  the  multiplicand  for  the  product;  consequently,  mul- 
tiplying one  fraction  by  another,  must  produce  a  fraction 
smaller  than  either  of  the  factors.  For  example,  -^^  mul- 
tiplied by  -^Q  is  -f^Q ,  or,  decimally,  .9  multiplied  by  .8  is 
.72.  Hence  you  may  observe,  that  the  number  of  decimal 
figures  in  any  product,  must  be  equal  to  the  number  of 
decimal  figures  in  both  the  factors. 

20.  Multiply  531  by  .52.  65.7  by  .43.  7.06  by  .24. 
.439  by  .38.     .149  by  .26. 


531 

65.7 

7.06 

.439 

.149 

.52 

.4  3 

.24 

.38 

.26 

1062 

1971 

2824 

3512 

894 

2655 

2628 

1412 

1317 

298 

276.12 

28.251 

1.6944 

.16682 

.03874 

186  SUPPLEMENT.  IL 

RULE  FOR  MULTIPLICATION  OF  DECIMALS.  Mul- 
tiply as  in  whole  numbers ;  and  in  the  product,  point  off 
as  many  figures  for  decimals,  as  there  are  decimal  places 
in  both  factors.  If  the  number  of  figures  in  the  product 
be  less  than  the  number  of  decimal  places  in  both  factors, 
prefix  ciphers  to  supply  the  deficiency, 

21.  Multiply  1608  by  .4,— that  is,  find  .4  of  1608. 

22.  Multiply  .45  of  a  dollar  by  8. 

23.  How  much  is  36  times  .495  of  a  dollar  ? 

24.  What  cost  18  yards  of  cloth,  at  $4,072  per  yd.? 

25.  What  cost  28.7  yards  of  cloth,  at  $  9  per  yd.  ? 

26.  What  cost  9.3  acres  of  land,  at  $8.41  per  acre? 

27.  If  1  yard  of  silk  cord  cost  7  mills,  [.007],  what  is 
the  price  of  .9  of  a  yard  ? 

.     28.  What  is  6  per  cent,  or  .06  of  340.4? 

29.  Multiply  42.863  by  70.28. 

30.  Multiply  2046  by  .932. 

31.  Multiply  .7253  by  .0423. 

32.  Multiply  6.5431  by  .402. 

33.  What  is  the  product  of  .04  multiplied  by  .07? 

34.  What  is  the  product  of  .005  by  .009  ? 

35.  Multiply  7  and  t>-hundredths  by  6-thousandths. 

DIVISION    OF    DECIMALS. 

RULE  FOR  DIVISION  OF  DECIMALS.  Divide  as  in 
whole  numbers ;  and  in  the  quotient,  point  off  as  many 
figures  for  decimals,  as  the  decimal  places  in  the  dividend 
exceed  those  in  the  divisor;  that  is,  make  the  decimal 
places  in  the  divisor  and  quotient  counted  together,  equal 
to  the  decimal  places  in  the  dividend. 

If  there  be  not  figures  enough  in  the  quotient  to  point 
off,  prefix  ciphers  *o  supply  the  deficiency. 

When  there  are  more  decimal  places  in  the  divisor, 
than  in  the  dividend,  render  the  places  equal,  by  annex^ 
ing  ciphers  to  the  dividend,  before  dividing. 

After  dividing  all  the  figures  in  the  dividend,  if  there 
be  a  remainder,  ciphers  may  be  annexed  to  it,  and  the 
division  continued.  The  ciphers  thus  annexed,  must  be 
counted  with  the  decimal  places  of  the  dividend. 


II.  DECIMALS.  187 

36.  Divide  64.395  by  40.5.     Divide  5.8674  by  127. 
40.5)64.395(1.59         127)  5.8674  (.0462 
405  508 

2389  787 

2025  762 


3645  254 

3645  254 

37.  Divide  2033.1  by  .324.  Divide  1383.2  by  60.8. 

.324)2033.100(6275  60.8)1383.2(22.75 

1944  1216 

891  1672 

648  1216 


2430  4560 

2268  4256 


1620  3040 

1620  3040 


38.  How  many  times  is  .27  contained  in  1.224? 

The    sign    of    addition,    or 
.27)  1.224(4.533+    more,  here  shows,  that  the  true 
108  quotient  is  more  than  the  pre- 

244  ceding    figures   express.       We 

jgf^  might    continually    annex    ci- 

— -—  phers   to   this    remainder,   and 

^^  carry  on  the  division,  but  we 

should  never  arrive  at  a  com- 
90  plete  quotient.     For  the  pur- 

81  poses  of  business,  it  is  seldom 

9  necessary  to   extend  the  quo- 

tient below  thousandths. 

39.  How  many  times  is  1.23  contained  in  3021.741  ? 

40.  How  many  times  is  1243.4  contained  in  5.37148? 

41.  How  many  times  is  .204  contained  in  77112? 

42.  How  many  times  is  4.2  contained  in  194.334  ? 

43.  How  many  times  is  30.02  contained  in  94.657  ? 

44.  How  many  times  is  .44  contained  in  .1606? 

45.  What  is  the  quotient  of  42.65  divided  by  36  ? 

46.  What  is  the  quotient  of  .8  divided  by  8  ? 


188  SUPPLEMENT.  H, 

CHANGE  OF  COMMON  FRACTIONS  TO  DECIMALS. 

RULE.  Annex  ciphers  to  the  numerator,  and  divide 
it  by  the  denominator :  the  quotient  will  be  the  decimal. 

47.  Change  -^-^  to  a  decimal. 

By  annexing  four  ciphers,  we  ob- 
12) 7.0000         ^^\j^  fQuj.  decimal  figures.     We  might, 
.5833+     however,   annex    more  ciphers,   and 
carry  the  decimal  still  lower. 

48.  Change  ^  to  a  decimal. 

49.  Change  the  following  fractions  to  their  respective 

Hppimak       2       1        3        13         5        _5_       14         26 
aecimais.  .3-.    4.    4.     ^o  •    TF'    12'    TT'    ^Yl' 

50.  Change  -j^^  of  a  dollar  to  a  decimal ;  that  is,  find 
how  many  cents  and  mills  there  are  in  ^  of  a  dollar. 

51.  Change  $48|  to  a  decimal  expression. 

52.  Change  <^316-^|  to  a  decimal  expression. 

CHANGE  OF  COMPOUND  NUMBERS  TO  DECIMALS. 

To  reduce  the  lower  denominations  of  a  compound 
number  to  the  decimal  of  a  higher  .denomination. 

RULE.  Reduce  the  given  quantity  to  a  common  fraction, 
then  change  this  fraction  to  a  decimal.     See  page  171. 

53.  Reduce  7s.  6d.  to  the  decimal  of  a  £. 

54.  Reduce  15  shillings  to  the  decimal  of  a  £. 

55.  Reduce  6d.  3qr.  to  the  decimal  of  a  shilling. 

56.  Reduce  2s.  lid.  3qr,  to  the  decimal  of  a  £. 

57.  Reduce  1  farthing  to  the  decimal  of  a  shiUing. 

58.  Reduce  £  18  2s.  7d.  to  a  decimal  expression. 

59.  Reduce  I4dwt.  18gr.  to  the  decimal  of  an  oz.Troy 

60.  Reduce  4qt.  Ipt.  to  the  decimal  of  a  bushel. 

61.  Reduce  3qt.  Ipt.  2gl.  to  the  decimal  of  a  gallon. 

62.  Reduce  lOr.  3yd.  2ft.  to  the  decimal  of  a  mile. 

63.  Express  29yd.  2qr.  3na.  of  cloth  decimally,  and 
find  its  cost,  at  $7,625  per  yard. 

CHANGE  OF  DECIMALS  TO  COMPOUND  NUMBERS. 

To  reduce  the  decimal,  of  a  higher  denomination,  to  its 
value  in  whole  numbers  of  lower  denomination. 

RULE.     Multiply  the  decimal  by  that  number  of  the  next 


Hi  DECIMALS.  109 

lower  denomination,  which  makes  a  unit  of  the  higher^  and 
the  product  will  be  of  the  lower  denomination.  Proceed 
thus  with  the  decimal  in  each  succeeding  product. 

64.  Reduce  .6526  of  a  .£  to  its  value  in  shillings,  &c. 
.6526  We   multiply   the   decimal  of   a  ^ 

20         by  20,  to   find   the   shillings,  because, 

13.0520         there    are    20     times    more    shillings 

J  2         than  pounds  in  any  sum,  whether  the 

^         sum  be  a  whole  number  or  a  decimal. 

.0^40         rp^g   g^jjjg   reasoning   also    applies,   in 

Z         finding   the   pence,  and   the   farthings. 

2.4960         Answer,  13s.  Od.  2qr.+ 

65.  Reduce  .4039  of  a  <£  to  its  value  in  shillings,  &c. 
QQ.  Reduce  .857  of  a  shilling  to  pence  and  farthings. 

67.  Reduce  .76  of  a  ton  to  cwt.  qr.  lb.  &;c. 

68.  In  .2094  of  a  day,  how  many  hours,  minutes,  &c.  ? 

69.  In  .57  of  an  acre,  how  many  roods,  rods,  &c.  ? 

70.  Reduce  .£15.2908  to  its  proper  expression  in 
pounds,  shillings,  pence,  and  farthings. 

EXCHANGE  OF  CURRENCIES. 

In  New  England,  Virginia,  Kentucky,  and  Tennessee, 
■|  of  a  dollar  is  called  a  shilling. 

In  New  York  and  North  Carolina,  |  of  a  dollar  is 
called  a  shilling. 

In  Pennsylvania,  New  Jersey,  Delaware,  and  Maryland, 
j^y  of  a  dollar  is  called  a  shilling. 

In  South  Carolina  and  Georgia,  -j\  of  a  dollar  is 
called  a  shilling. 

In  Canada,  |  of  a  dollar  is  called  a  shilling. 

In  Great  Britain,  the  shilling,  of  the  Sterling  currency, 
IS  equal  to  f  of  a  dollar. 

71.  How  many  cents  and  mills,  that  is,  what  decimal 
of  a  dollar,  in  a  New  England  shilling  ?  in  2  shillings  ? 
in  3  shillings?    in  4  shillings?   in  5  shillings? 

6)  1.000       6) 2.000        ,  .i^f  ^^^  """^^^^  expressing 

ifi^  qoqi     shilhngs,  expresses  an  equal 

•ioo^  .666-^    value  in  decimals  of  a  dollar. 


190  SUPPLEMEJNT.  It. 

72.  How  many  cents  and  mills  in  a  New  York  shil- 
ling ?  in  2s.  ?    in  3s.  ?    in  4s.  ?    in  5s.  ?    in  6s.  ?    in  7s.  ? 

73.  How  many  cents  and  mills  in  a  Pennsylvania  shil- 
ling ?  in  2s.  ?     in  3s.  ?     in  4s.  ?     in  5s.  ?     in  6s.  ? 

lX^=.133i      '2xf5  =  .266| 

74.  How  many  cents  and  mills  in  a  Georgia  shilling? 
in  2s.  ?     in  3s.  ?     in  4s.  ? 

75.  How  many  cents  and  mills  in  a  Canada  shilling? 
in  2s,  ?     in  3s.  ?     in  4s.  ?     in  5s.  ? 

76.  How  many  cents  and  mills  in  a  shilling,  Sterling, 
of  Great  Britain  ?     in  2s.  ?     in  3s.  ?     in  4s.  ? 

To  change  the  currencies  of  pounds,  shillings  and  pence, 
of  every  variety  of  value,  to  Federal  money. 

RULE.  Reduce  the  pounds,  if  there  be  any,  to  shillings. 
Denote  the  shillings  as  units,  reduce  the  pence  and  far- 
things to  the  decimal  of  a  shilling,  and  multiply  the  sum  by 
that  fraction  of  a  dollar  which  is  equal  to  one  shilling. 

77.  Change  13s.  6d.,  of  the  old  currency  of  New  Eng- 
land, to  Federal  money. 

13s.  6d.  =  13.5s.         13.5  X  i  =  2.25. 

78.  Change  ,£42  19s.  4^d.,  of  the  old  currency  of  New 
England,  to  Federal  money. 

£42  19s.  4  id.  =  859.3758. 

79.  Change  13s.  6d.,  of  the  old  currency  of  New  York, 
to  Federal  money. 

80.  Change  £  25  17s.  8 1  d.,  of  the  old  currency  of  New 
York,  to  Federal  money. 

81.  Change  18s.  lid.,  of  the  old  currency  of  Pennsyl- 
vania, to  Federal  money. 

82.  Change  ,£  14  7s.  6  ^  d.,  of  the  old  currency  of  Penn- 
sylvania, to  ^cjderal  money. 

83.  Change  16s.  lOd.,  of  the  old  currency  of  Georgia, 
to  Federal  money. 

84.  Change  .£54  12s.  l^d.,  of  the  old  currency  of 
Georgia,  to  Federal  money. 

85.  Change  £21  9s.  3|d.,  of  the  currency  of  Canada, 
to  Federal  money. 

86.  Change  ,£  5  12s.  4d.  Sterling,  of  Great  Britain,  to 
Federal  money. 


in  PERCENTAGE.  191 

III. 

PERCENTAGE. 

Percentage  has  already  been  explained  iii  page  163. 
Since  per  cent,  indicates  hundredths,  it  is  properly  ex- 
pressed in  the  first  and  second  decimal  places,  taken  to- 
gether. Thus,  6  percent,  is  .06  ;  12  percent,  is  .12.  A 
fraction  of  1  per  cent,  is  expressed  in  decimals  lower  than 
hundredths.  Thus,  ^  per  cent,  is  .005 ;  \  per  cent,  is 
.0025  ;  6^  per  cent,  is  .065;  12 1  per  cent,  is  .1275. 

Multiplying  by  a  decimal,  produces  such  a  part  of  the 
multiplicand,  as  the  decimal  indicates.     Therefore, — 

To  FIND  THE  Percentage  on  any  sum, — Multiply 
the  sum  by  the  decimal  which  denotes  the  rate  per  cent, 

1.  A  merchant  having  $1426  in  the  bank,  drew  out  5 
per  cent,  of  it.     What  sum  did  he  draw  ? 

14  26         Since  5  per  cent,  of  any  quantity  is  -j^q  of 
.0  5     that  quantity,  the  question  in  this  example  is, 
t;l  ,  on     ^^^^^t  is  -j§^  of  1426  dollars  ?    Or,  decimally, 
^^  ^"^^     What  is  .05  of  1426  dollars  ? 

2.  V/hat  is  1  per  cent,  of  $  100?     of  $834  ? 

3.  What  is  3  per  cent,  of  $  100  ?     of  $42? 

4.  What  is  7  per  cent,  of  $  100  ?     of  $  1085  ? 

5.  What  is  9  per  cent,  of  354  dollars  ? 

6.  What  is  24  per  cent,  of  1852  dolIai*s? 

When  the  rate  is  a  ii-action  of  1  per  cent.  —  First,  re- 
duce the  rate  to  a  decimal,  by  multiplying  .01  by  the  frac* 
tion.     Then  multiply  by  the  decimal  rate  as  before. 

7.  What  is  |  per  cent,  of  234  dollars  ? 

.01 X  I— .0075.     Then  234x  .0075 nn  1.755. 

8.  What  is  \  per  cent,  of  524  dollars  ? 

9.  What  is  |  per  cent,  of  190  dollars? 

10.  What  is  2i  per  cent.  [.025]  of  50  dollars? 

11.  What  is  6i  per  cent,  of  75  dollars? 

12.  What  is  10 1  per  cent,  of  2C0  dollars? 


192  SUPPLEMENT.  Ill 

13.  Find  7^  per  cent,  of  344  dolls. 

When  there  is  a  fraction  in  the     3444-100=3.44 
rate  per  cent,  which  cannot  be  ex-  7^ 

actly   expressed   by  a   decimal^— as  ^408 

in  this  example — we  first  find  1  per  114-^ 

cent,  of  the   given  sum,  by  divid-  ¥qTo"o2 

ing  it  by  100,  and  then  multiply  this  ^x2  0.^^3 

quotient  by  the  mixed  number  that  expresses  the  rate. 

14.  What  is  4 1  per  cent,  of  624  dollars  ? 

15.  What  is  6f  per  cent,  of  38  dollars  ? 

16.  What  is  3i  per  cent,  of  2310  dollars? 

17.  What  is  9|  per  cent,  of  17  dollars? 

18.  What  is  7  per  cent,  of  24  dolls.  32  cts.  ? 

Here  we  have  cents  [decimals]  in  the  num-  24  3  2 

ber  on  which  the  percentage  is  to  be  taken.  *q^ 

We  however  multiply  as  usual  in  decimals,     r — 

and  the  first  two  decimal  figures  in  the  prod-  ^  ^  '*  ^z^"* 
uct  express  cents,  the  third  mills,  the  fourth  tenths  of  ^.  mill* 

19.  What  is  14  percent,  of  S641.94? 

20.  What  is  4}  percent,  of  $37.26? 

21.  What  is  11 -J- per  cent,  of  1 150.75  ? 

To  find  what  per  cent.*a  smaller  number  is  of  a  larger,—* 
Consider  the  smaller  number  as  a  numerator,  and  the 
larger  as  a  denominator  of  a  fraction ;  then  reduce 
this  fraction  to  a  decimal.     See  page  188. 

22.  If  a  man,  having  $  94  deposited  in  bank,  draw  out 
$  25,  what  per  cent,  of  his  deposit  does  he  draw  ? 

25  is  fl  of  94.  Thenf|z=.26f|-.   Ans.  26  |f  per  cent. 

23    What  per  cent,  of  240  dollars  is  32  dollars? 

24.  What  per  cent,  of  12  dollars  is  7  dollars  ? 

25.  What  per  cent,  of  $95.21  (9521  cts.),  is  §4.22? 

To  find  a  percentage  of  a  compound  number, — Multi- 
ply by  the  rate  per  cent.,  as  a  ivhole,  or  mixed  number,  and 
divide  the  product  by  100,  or  the  factors  of  100. 

26.  What  is  6  per  cent,  of  ^22  10s.  9d.  ? 

27.  What  is  4  per  cent,  of  .£41  15s.  6d.  ? 

28.  What  is  3^  per  cent,  of  £8  16s.  8d.  ? 


Ill  PERCENTAGE:  193 

COMMISSION. 

Commission  is  the  compensation  made  to  factors  and 
brokers  for  their  services  in  buying  or  selling.  It  is  rec- 
koned at  so  much  per  cent,  on  the  money  employed  in 
the  transaction. 

29.  What  is  the  commission  on  8500,  at  2-^  per  cent.  ? 

30.  If  I  allow  my  factor  a  commission  of  3  per  cent,  for 
disbursing  725  dollars  50  cents,  on  my  account,  what  does 
his  commission  amount  to  ? 

3 1 .  How  much  does  a  broker  receive  for  his  services  on 
a  sale  of  stocks  amounting  to  52648  dollars,  allowing  his 
commission  to  be  ^  of  1  per  cent.  ? 

STOCKS. 

Stock  is  a  property,  consisting  in  shares  of  some  estab- 
lishment, designed  to  yield  an  income.  It  includes  gov- 
ernment securities,  shares  in  incorporated  banks,  insurance 
offices,  factories,  canals,  rail-roads,  &c. 

The  par  value  of  a  share,  is  what  it  originally  cost ;  and 
the  real  value,  at  any  time,  is  what  it  can  be  sold  for* 
When  it  will  sell  for  more  than  it  originally  cost,  it  is  said 
to  be  above  par,  and  the  excess  is  stated  at  so  much 
per  cent,  advance.  When  its  real  value  is  less  than  the 
original  cost,  it  is  below  par,  and  is  sold  at  a  discount. 

32.  Sold  10  shares  in  the  Manufacturers'  Insurance 
Company,  at  5  per  cent,  advance,  the  par  value  of  a  share 
being  100  dollars.     How  much  did  I  receive? 

33.  Bought  15  shares  in  the  Boston  Bank,  at  |  of  1 
per  cent,  advance,  the  par  value  being  50  dollars  a  share. 
How  much  did  I  give  for  them  ? 

34.  Sold  64  shares  in  the  State  Rail-road,  at  \\  per 
cent,  discount,  the  par  value  being  100  dollars  a  share. 
How  much  did  I  receive  for  them  ? 

INSURANCE. 

Insurance  is  security  given,  to  restore  the  value  o' 
ships,  houses,  goods,  ^c,  which  may  be  lost  at  sea,  c 

R 


194  SUPPLEMENT.  lY 

by  fire.  The  security  is  given  in  consideration  of  a  j^rc 
mium  paid  by  the  owner  of  the  property  insured.  This 
pretnium  is  a  percentage  on  the  value  of  the  property. 

The  written  instrument,  which  is  the  evidence  of  the 
contract  of  indemnity,  is  called  a  policy. 

35.  What  is  the  amount  of  premium  for  insuring  19416 
dollars,  at  2  -^-  per  cent.,  on  a  ship  from  Liverpool  ? 

36.  I  effected  an  insurance  of  3460  dollars  on  my 
dwelling-house  for  one  year,  at  |  of  1  per  cent.  What 
did  the  premium  amount  to? 

37.  If  you  obtain  an  insurance  on  goods  valued  at  $7325, 
at  i  of  I  per  ceiit.,  what  will  the  premium  amount  to  ? 


iv. 

INTEREST. 

Interest  has  already  been  defined,  and  rules  f6r  com-^ 
puting  it  without  decimals  have  been  given,  in  Chap.  VI., 
Sect.  23.  The  rules  are  repeated  in  this  article,  with 
such  modifications  as  provide  for  the  use  of  decimals. 

To  compute  interest  for  one  or  more  years. 

RULE.  Multiply  the  principal  by  the  decimal  that  ex-^ 
presses  the  rate,  and  the  product  will  he  the  interest  for 
1  year.  Multiply  the  interest  for  one  year  by  the  number 
of  years o 

1.  Find  the  interest  of  $  87.41  j  for  3  years,  at  6  per  cent* 
87.41  X  .06  X  3=15.7338.  Ans.  $  15.73+ 

In  the  answers,  fractions  of  a  cent,  may  be  omitted. 

2.  Find  the  interest  of  f  644,^  for  4  years,  at  6  per  cent, 

3.  Find  the  interest  of  92  cents,  for  7  years,  at  6  percent. 

4.  Find  the  interest  of  f  7.50,  for  2  years,  at  4  per  cent* 

5.  Find  the  interest  of  i  2.91,  for  3  yeai^,  at  4  ^  percent. 

6.  Find  the  interest  of  $  9.53,  for  4  years,  at  5 1  per  cent. 

7.  What  is  the  interest  of  $752.25,  for  3  years,  at  5 
per  cent.  ?     What  is  the  amount  ? 

8.  What  is  the  interest  of  £  16  8s.  6d.,  for  1  year,  at  6 
per  cent.  ?     What  is  the  amount  ? 


iV.  INTEREST.  19(5 

To  compute  interest  when  there  are  months  in  the  time. 

RULE.  First  find  the  interest  for  the  years ,  if  there 
be  any.  Then  take  ^^^  of  a  yearns  interest  for  1  month ; 
T2  ^^  6  fi^  ^  months ;  -^^  or  \  for  3  months ;  and  so  on, 

9.  What  is  the  interest  of  224  dollars  for  7  months,  at 
6  per  cent,  per  annum  ? 

10.  What  is  the  interest  of  75  dollars  and  50  cents,  for 
5  months,  at  6  per  cent.  ? 

11.  What  is  the  interest  of  145  dollars,  for  1  year  and 
3  months,  at  6  per  cent.  ? 

12.  What  is  the  interest  of  95  dollars  and  25  cents,  for 
2  years  and  8  months,  at  5  per  cent.  ? 

13.  What  is  the  interest  of  $351.09,  for  3  years  and  9 
months,  at  7  per  cent.  ?     What  is  the  amount  ? 

To  compute  interest,  when  there  are  days  in  the  time. 

RULE.  First  find  the  interest  for  the  years  and  months , 
if  there  he  any.  Then  take  -^^  of  a  month's  interest  for  1 
day ;  f^  or  ^-^  for  2  days ;  -^^  or  -^^  for  3  days ;  and  so  on. 

14.  What  is  the  interest  of  $  1000  for  1  year,  1  month 
and  1  day,  at  6  per  cent.  ? 

15.  What  is  the  interest  of  $356.75  for  8  months  and 
10  days,  at  6  per  cent.  ? 

16.  What  is  the  interest  of  $76.81  for  5  years,  2*  months 
and  18  days,  at  4  per  cent.  ? 

17.  What  is  the  interest  of  $  250  for  1  year  and  29 
days,  at  6  per  cent.  ?     What  is  the  amount  ? 

18.  What  is  the  amount  of  $92.86  for  3  years,  7 
months  and  14  days,  at  7  per  cent.  ? 

19.  What  is  the  interest  of  $  175.63,  from  May  19, 
1842,  to  January  4,  1844,  at  6  per  cent.  ? 

We  find  the  time  between  the  two  1844  1  4 
dates  by  subtracting  the  first  from  the  1842  5  19 
last,  as  in  compound  subtraction  ;  the  7  YTE 
months  being  denoted  numerically.  

20.  What  is  the  interest  of  $208.90,  from  June  2, 1843, 
to  August  4,  1845,  at  5i  per  cent.  ? 

21.  What  is  the  interest,  at  6  per  cent.,  on  a  note  of 
$110,  dated  Sept.  7,  1843,  and  paid  July  9,  1846? 
What  is  the  amount  ? 


196  SUPPLEMENT.  IV 

PARTIAL  PAYMENTS. 

In  the  settlement  of  notes,  which  have  been  partly  paid, 
at  dates  previous  to  the  settlement,  the  common  method  of 
computing  the  interest  operates  justly  in  cases  where  inter- 
est has  not  been  running  for  more  thtm  one  year.  But  for 
longer  periods  of  interest,  this  method  is  not  strictly  just  to 
the  creditor,  and  ought  not  to  be  adopted. 

THE  COMMON  METHOD.  Compute  the  interest  on  the 
original  debt,  from  the  date  when  interest  commenced  to 
the  date  of  the  settlement ;  also,  compute  the  interest  on 
each  payment,  from  the  date  of  the  payment,  to  the  date 
of  the  settlement.  Then,  subtract  the  amount  of  all  the 
payments  from  the  amount  of  the  original  debt,  and  the 
remainder  will  be  the  balance  due. 

The  United  States'  Court,  and  the  Courts  of  the  several 
States,  in  which  decisions  have  been  reported  —  with  the 
exception  of  Connecticut,  Vermont,  and  New  Jersey — 
have  established  a  uniform  legal  rule  for  the  computation 
of  interest,  when  partial  payments  have  been  made. 

THE  LEGAL  RULE.  Compute  the  interest  on  the 
principal  of  the  note  to  the  earliest  date  when  a  payment 
was  made,  which,  either  alone,  or  together  with  preceding 
payments,  exceeds  the  interest  then  due.  Add  this  inter^ 
est  to  the  principal,  and  from  the  sum.  subtract  the  pay- 
ment or  payments  thus  far  made.  The  remainder  be- 
comes a  new  principal,  with  which  proceed  as  with  the 
principal  of  the  note. 

(22.)  Boston,  January  14th,  1843. 

For  value  received^  I  promise  to  pay  Wm.  Rich,  or  or- 
der, one  hundred  and  forty-one  dollars  and  eight  cents,  in 
three  months,  with  interest  after.  John  Lang. 

On  the  back  of  this  note  were  the  following  endorsements.  May 
1st,  1843,  received  seventy-five  dollars.  September  14th,  1843,  re- 
ceived forty-five  dollars.  What  is  the  balance  Jan.  14th,  1844,  the 
interest  being  6  per  cent.,  computed  by  the  common  method? 


First  payment,     $  75. 
Int,  8  m.  14  d.,        3.17 


Sndpayt.  $45. 
Int.,  4  m.         .90 


Amount,  $78.17    Amount,  $45.90 

78.17 

Amount  of  payments,         $124.07 


Principal,      $  141.08 
Int.,  9  m.  6.34 

Amount,  147.42 

124.07 

Balance,         $2a35 


IV.  INTEREST.  197 

(23.)  New  York,  May  25th,  1843. 

For  value  received,  I  promise  Joseph  Day  to  pay  him 
or  order,  the  sum  of  three  hundred  and  one  dollars  and 
forty-seven  cents,  on  demand,  with  interest. 

Attest,  John  Smith.  Samuel  French. 

On  the  back  of  this  note,  the  following  endorsements  were  made. 
July  1st,  1843,  received  sixty-seven  dollars  and  fifty  cents.  Janu- 
ary 4th,  1844,  received  forty-eight  dollars.  April  11th,  1844,  re- 
ceived thirty-nine  dollars.  What  is  the  balance,  June  21st,  1844 ; 
interest  being  6  per  cent,  computed  by  the  common  method  ? 

(24.)  Philadelphia,  March  4th,  1842. 

For  value  received,  I  promise  to  pay  to  the  order  of 
Harper  h  Jones,  one  thousand  two  hundred  dollars,  on 
demand,  with  interest.  Charles  Train. 

The  following  endorsements  are  on  this  note.  June  10th,  1842, 
received  one  hundred  sixty-nine  dollars  and  twenty  cents.  Oct 
22d,  1842,  received  twenty  dollars.  March  30th,  1843,  received 
twenty-eight  dollars.,  Nov.  5tli,  1843,  received  six  hundred  eigh- 
teen dollars  and  five  cents.  If  6  per  cent  interest  be  computed  by 
the  legal  rule,  what  is  tlie  balance  due,  March  5th,  1844  ? 

Principal,  -  $1200. 

Interest  from  Mar.  4,  to  June  10,  (3  m.  6  d.),  -        -        19.20 

First  Amount,  -        -    1219.20 

First  payment,  169.20 

Balance,  forming  a  new  principal,          -        _  .        .    1050.00 

Interest  from  June  10,  to  Oct  22,  (4  m.  12  d.),  $23.10 

Second  payment, 20. 

Leaving  interest  unpaid,        -        .        -        _  3.10 

Interest  from  Oct  22,  to  Mar.  30,  (5m.  8  d.),  27.65 

30.75 

Third  payment,  28.00 

Leaving  interest  unpaid,        _        -        -        -  2.75 

Interest  from  Mar.  30,  to  Nov.  5,  (7  m.  6  d.),  37.80        40.55 

Second  Amount,  -  -  1090.55 

Fourth  payment,           -        -        -        -        -  -  -  618.05 

Balance,  forming  a  new  principal,          -        -  -  -  472.50 

Interest  from  Nov.  5,  to  Mar.  5,  (4  m.),           -  -  -  9.45 

Balance  due  on  taking  up  the  note,       -        -  -  $  481.95 

(25.)  New  Orleans,  January  1,  1841. 

For  value  received,  I  promise  to  pay  William   Lock 

or  order,  one  thousand  dollars,'  on  demand,  with  interest, 
6  per  cent.  Edward  Smith. 


19S  SUPPLEMEISIT.  IV 

Five  partial  payments  are  endorsed  on  Smith's  note :  viz.  Fe^. 
Ist,  1842,  received  seventy- five  dollars.  June  1st,  1842,  received 
twenty  dollars.  August  1st,  1843,  received  twenty  dollars.  October 
1st,  1843,  received  seven  hundred  and  fifty  dollars.  Feb.  1st  1844, 
received  one  hundred  dollars.  The  balance  of  this  note  was  paid 
June  1st,  1844.    How  much  was  it,  by  the  legal  rule  ? 


COMPOUND  INTEREST. 

Compound  Interest  is  that  which  is  paid  not  only  for 
the  use  of  the  principal,  but  also  for  the  use  of  the  interest 
after  it  becomes  due.  The  period  of  interest,  that  is,  the 
term  of  time  at  the  end  of  which  interest  is  due,  may  be  a 
year,  a  quarter,  or  any  other  term  agreed  upon.  What- 
ever be  the  period,  the  following  rule  is  applicable. 

RULE.  Find  the  amount  for  the  first  period^  and 
consider  it  the  principal  for  the  second  period;  find  the 
amount  for  the  second  period,  and  consider  it  the  prin- 
cipal for  the  third  period ;  and.  thus  proceed  through  the 
whole  number  of  periods.  Subtract  the  first  principal 
from  the  last  amount,  and  the  remainder  will  be  the 
compound  interest, 

26.  What  is  the  compound  interest  of  $  100  for  3  years, 
at  6  per  cent. ;  the  interest  being  due  annually  ? 

1st  year.  2nd  year.  3rd  year.  Answer. 

100     106     112.36    119.10  + 
.06      .06     ^06    100 

6.00     6.36     6.7416    $19.10  + 
100     106      112.36      

106.00   112.36    119.1016 

27.  Wliat  is  the  compound  interest  of  355  dollars,  for  6 
years,  at  6  per  cent,  per  annum  ? 

28.  What  is  the  compound  interest  of  250  dollars,  for  4 
years,  at  7  per  cent,  per  annum  ? 

29.  To  what  sum  will  450  dollars  amount,  in  5  years, 
at  5  per  cent,  per  annum,  compound  interest  ? 

30.  At  compound  interest,  what  will  600  dollars  amount 
to  in  1  i  year,  at  the  rate  of  6  per  cent,  a  year,  interest 
payable  quarterly  ? 


V.   VI  BANKING.  199 


DISCOUNT. 

Discount  is  sufficiently  defined  in  page  168;  and  we 
have  now  only  to  apply  decimals  to  the  operations. 

RULE.  Divide  the  debt  by  the  amount  of  1  dollar  for 
the  time,  and,  the  quotient  is  the  present  worth.  Subtract 
the  present  worth  from  the  debt,  and  the  remainder  ivill 
be  the  discount, 

1.  What  is  the  present  worth  of  125  dollars,  due  in  18 
months,  when  interest  is  6  per  cent,  per  annum  ? 

$1  amounts  to  $1.09.        125^1.09=114.66+ 

2.  What  is  the  present  worth  of  $  456,  due  in  15  months, 
when  money  is  worth  5  per  cent,  per  annum  ? 

3.  What  is  the  discount  on  3465  dollars  for  6  months, 
when  interest  is  7  per  cent,  a  year  ? 

4.  What  is  the  present  value  of  a  note  for  2448  dollars 
and  50  cents,  payable  in  8  months,  when  interest  is  6  per 
cent,  per  annum  ? 


VI. 
BANKING. 

The  interest  on  money  hired  from  a  bank  is  paid  when 
the  money  is  taken  out.  That  is,  the  bank  computes  the 
interest  on  the  principal  of  the  note  it  receives,  to  the  time 
the  note  is  to  be  paid,  deducts  this  interest  from  the  princi- 
pal, and  advances  the  remainder  to  the  hirer.  Hence, bank 
mterest  is  called  discount;  and  the  note  received  by  the 
bank  is  said  to  be  discounted. 

Bank   discount  is  always  computed   for  three  days  — 
Q^Wedi  days  of  grace  —  more  than  the  time  stated  in  the 
note  for  payment;  and  the  hirer  is  not  required  to  pay. 
until  the  last  of  these  three  days. 

1.  Find  the  bank  discount  on  $585  for  60  days  and 
grace.  (2^^^  months,)  at  the  rate  of  6  per  cent,  a  year 


200  SUPPLEMENT.  VH 

2.  What  is  the  bank  discount  on  900  dollars  for  90  days, 
and  grace,  at  the  rate  of  6  per  cent,  a  year  ? 

3.  How  much  is  received  on  a  note  for  2540  dollars  80 
cents,  payable  in  4  months,  and  grace,  discounted  at  a 
bank,  when  interest  is  4^  per  cent,  a  year  ? 

4.  A  note  for  452  dollars,  payable  in  7  months,  and 
grace,  is  discounted  at  a  bank,  when  interest  is  6  per  cent, 
per  annum.     What  sum  is  received  on  it  ? 


VIT. 
PROFIT   AND    LOSS. 

The  ascertaining  what  is  gained  or  lost  in  buying  and 
selling,  and  the  adjusting  of  the  price  of  goods  so  as  to 
gain  or  lose  a  certain  sum,  or  a  certain  per  cent.,  come  un- 
der the  head  of  Profit  and  Loss, 

1.  Bought  a  piece  of  broadcloth  containing  28  yards  for 
112  dollars,  and  sold  it  at  5  dollars  25  cents  a  yard.  How 
much,  and  what  per  cent.,  was  my  profit  ?  (See  Percent- 
age, Art.  III.,  Example  22.) 

2.  Bought  3  pieces  of  broadcloth,  containing  28  yards 
each,  at  5  dollars  25  cents  a  yard.  At  what  price  per 
yard  must  I  sell  it,  to  gain  20  per  cent.  ? 

3.  Bought  cloth  at  4  dollars  60  cents  a  yard,  which, 
not  proving  so  good  as  I  expected,  I  sold  at  3  dollars  91 
cents  a  yard.     What  per  cent,  did  I  lose  ? 

4.  Bought  1250  barrels  of  flour  for  6250  dollars.  At 
what  price  per  barrel  must  I  sell  it,  to  make  a  profit  of 
12^-  percent.? 

5.  Bought  wheat  at  75  cents  a  bushel ;  at  what  price 
per  bushel  must  I  sell  it,  to  gain  20  per  cent.  ? 

6.  A  merchant  received  from  Lisbon  180  casks  of  rai- 
sins, containing  80|-lb.  each,  which  cost  him  2  dollars  18 
cents  a  cask.  At  what  price  per  cwt.  must  he  sell  them, 
to  gain  25  per  cent.  ? 

7.  If  I  sell  sugar  at  8  dollars  per  cwt.,  and  thereby 
lose  12  per  cent.,  what  per  cent,  do  I  gain  or  lose,  bv 
selling  the  same  at  9  dollars  per  cwt.  ? 


VIII  PARTNERSHIP.  201 

VIIL 

PARTNERSHIP. 

Partnership  is  the  union  of  two  or  more  individuals  in 
trade.  The  company  thus  associated  is  called  a  firm  :  and 
the  amount  of  property,  which  each  partner  puts  into  the 
firm,  is  called  his  stock  in  trade. 

When  each  partner's  stock  is  in  the  firm  an  equal  length 
of  time,  the  profit  or  loss  is  shared  in  proportion  merely  to 
each  one's  stock.  But  when  the  stock  of  the  several  part^ 
ners  is  employed  unequal  terms  of  time,  the  profit  or  loss 
is  shared  in  proportion  both  to  stock  and  time, 

1.  A,  B,  and  C  entered  into  partnership,  and  the  stock 
of  each  was  in  the  firm  one  year.  A  put  in  240  dollars, 
B  360  dollars,  and  C  120  dollars.  They  gained  350 
dollars.     What  was  each  partner's  share  of  the  gain  ? 

Solution.  The  whole  capital  $720.  A's  stock  was 
$240,  and  he  must  have  f|§  of  the  gain.  B's  stock  was 
$360,  and  he  must  have  ||g.  C's  stock  was  $  120,  and 
he  must  have  ^|§.     Observe  the  following  statement. 

24  0 --2  .   and  f  of  $350.  is  116 f  dollars,  A's  share. 

3  60 -.3  .   and  I  of  $350.  is  175    dollars,  B's  share. 

120  =1-^;   and  |  of  $350.  is_58j-  dollars,  C's  share. 

$350   Proof. 

2.  W,  S,  and  L  formed  a  connexion  in  business. 
W  put  into  the  firm  2500  dollars ;  S  2000  dollars ;  and 
L  1500  dollars.  The  stock  of  the  several  partners  was 
m  trade  the  same  term  of  time,  and  they  gained  1500  dol- 
lars.   What  was  each  partner's  share  of  the  profit  ? 

3.  A,  B,  C,  and  D  traded  together  one  year.  A  put 
in  800  dollars,  B  500  dollars,  C  300  dollars,  and  D  150 
dollars ;  but  by  misfortune  they  lost  350  dollars.  What 
loss  did  each  partner  sustain  ? 

4.  Three  merchants  bought  a  ship,  for  8000  dollars. 
A  paid  2850  dollars,  B  1980  dollars,  and  C  the  rest.  In 
her  first  voyage  she  cleared  6400  dollars.  How  much  of 
the  profit  had  each  partner  ? 


202  SUPPLEMENT.  VIII. 

When  the  time  is  unequal,  we  compute  on  the  prin- 
ciple, that  $1  for  2  months  is  equal  to  $2  for  1  month. 
For  example,  A,  B,  and  C  traded  in  company ;  A  put  in 
$200  for  3  months,  B  $180  for  5  months,  and  C  $70 
for  10  months :  they  gained  $  132.  Now  we  say,  that 
A's  $200  for  3  months  was  the  same  as  $600  for  1 
month;  B's  $  ISO  for  5  months  the  same  as  $900  for  1 
month;  and  C's  $70  for  10  months  the  same  as  $700 
for  1  month ;  therefore  the  relation  is  the  same  as  if  A 
had  put  in  $600,  B  $900,  and  C  $700,  all  for  an  equal 
term  of  time.  These  sums  added  together  make  $2200; 
therefore,  A  had  ^6_o_o^  of  the  gain,  B  ^%%,  and  C  ^.}^. 
These  fractions,  reduced,  are  ^^,  -^-^^  and  ■^■^,  J^  of 
$132  is  $6;  then  A  had  6  times  $6,  B  9  times  $6, 
and  C  7  times   $6. 

RULE.  Multiply  each  partner^  stock  by  the  time  it 
was  in  the  firm ;  make  each  product  the  numerator  of  a 
fraction^  and  the  sum  of  the  products  a  common  denomi- 
nator ;  then  multiply  the  whole  gain  or  loss  by  each  of 
these  fractions,  for  each  partner^ s  share. 

5.  A,  B,  and  C  traded  in  company.  A  put  in  400 
dollars  for  9  months,  B  300  dollars  for  6  months,  and  C 
200  dollars  for  5  months :  they  gained  320  dollars. 
What  was  the  gain  of  each  ? 

6.  X,  Y,  and  Z  formed  a  partnership.  X  put  into 
the  firm  500  dollars  for  18  months,  Y  380  dollars  for  13 
months,  and  Z  270  dollars  for  9  months ;  but  they  lost 
818  dollars  50  cents.     What  was  the  loss  of  each  ? 

7.  Gould  and  Davis  entered  into  partnership  for  one 
year.  Gould's  stock,  at  first,  was  only  500  dollars,  but 
at  the  end  of  5  months  he  put  in  150  dollars  more. 
Davis's  stock,  at  first,  was  600  dollars,  but  at  the  end  of 
9  months  he  took  out  200  dollars :  at  the  end  of  the  year, 
it  was  found  they  had  gained  682  dollars  50  cents.  What 
was  the  gain  of  each  partner  ? 

8.  Three  farmers  hired  a  pasture  at  60  dollars  50 
cents  for  the  season.  A  put  in  5  cows  4^  months,  B  8 
cows  5  months,  and  C  9  cows  6^  months.  What  rent 
did  each  pav  ? 


IX, 


ASSESSMENT  OF  tAXES. 


203 


IX. 
ASSESSMENT  OF  TAXES. 

Taxes  are  imposts  paid  by  the  people  for  the  support 
of  government.  Tiiey  are  assessed  on  the  citizens  in 
proportion  to  their  property ;  except  the  poll  tax,  which 
is  assessed  by  the  head  without  regard  to  property. 

An  inventory  of  the  taxable  property  of  every  citizen 
is  the  first  thing  to  be  obtained  by  the  assessors. 

When  a  tax  is  on  property  and  polls,  we  deduct  the 
amount  which  the  polls  pay  from  the  sum  to  be  raised, 
and  apportion  the  remainder  according  to  each  man's 
property. 

To  effect  the  apportionment,  we  find  what  per  cent,  of 
the  whole  property  to  be  taxed,  the  sum  to  be  raised  is; 
and  if  we  multiply  each  man's  inventory  by  that  per 
cent,  expressed  in  decimals,  the  product  is  his  tax. 

Assessors  find  it  more  expedient,  however,  to  make  a 
table,  which  shall  exhibit  at  once  the  tax  on  all  sums, 
from  $  1  up  to  any  amount  required.  The  table  is  made 
by  multiplying  the  per  cent,  which  the  tax  amounts  to,  by 
the  several  numbers,  1,  2,  3,  4,  and  so  on. 

The  following  is  a  table  of  taxes  to  be  made  when  1^ 
per  cent,  is  to  be  raised  on  the  valuation  of  property. 


$1  pays 

.015 

$20 

pay   .30 

$200  pay 

$3.00 

2    " 

.03 

30 

"     .45 

300    " 

4.50 

3    " 

.045 

40 

"     .60 

400    " 

6.00 

4    " 

.06 

50 

«     .75 

500    " 

7.50 

5    « 

.075 

60 

"     .90 

600    " 

9.00 

6    « 

.09 

70 

"  1.05 

700    « 

10.50 

7    « 

.105 

80 

"  1.20 

800    " 

12.00 

8    " 

.12 

90 

"  1.35 

900    « 

13.50 

9    " 

.135 

loo 

"  1.50 

1000    " 

15.00 

10    " 

.15 

1.    By  the  al 

Jove  tabi 

e,  what  \v 

luld    be   the 

tax  on 

$6425  real  estate,  and  $2346  personal  estate  ? 

2.  By  the  above  table,  what  would  be  the  tax  of  a 
freeholder,  whose  real  estate  is  valued  at  $9842,  and 
personal  estate,  at  $  1 5066  ;  poll  tax  $  1.25  ? 


204  SUPPLEMENT  X. 

X. 

RATIO,   PROPORTION, 
RULE    OF    THREE. 

Ratio  Is  the  mutual  relation  of  two  numbers  to  one 
another.  By  finding  how  many  times  one  number  is  con 
tained  in  another,  or  what  part  one  number  is  of  another, 
we  obtain  their  ratio.  Thus,  the  ratio  of  2  to  4  is  2,  be- 
cause 2  is  contained  2  times  in  4 ;  and  the  inverse  ratio 
is  |,  because  2  is  |  of  4.  Both  these  expressions  of  the 
ratio  of  2  to  4  amount  to  the  same  thing,  which  is,  that 
one  of  the  numbers  is  twice  as  great  as  the  other. 

A  ratio  is  denoted  by  two  dots,  similar  to  a  colon  :  thus, 
3  :  9  expresses  the  ratio  of  3  to  9.  The  former  term  of  a 
ratio  is  called  the  antecedent,  and  the  latter  the  consequent. 
Thus,  6  :  12  expresses  the  ratio  of  6  to  12,  in  v/hich  6  is 
the  antecedent,  and  12  the  consequent. 

Since  a  ratio  indicates  how  many  times  one  number 
is  contained  in  another,  or  what  part  one  number  is  of 
another,  it  is  a  quotient,  resulting  from  the  division  of  one 
of  the  terms  of  the  ratio  by  the  other,  and  may  be  ex- 
pressed in  the  form  of  a  fraction  :  thus,  the  ratio  6  ;  3 
may  be  expressed,  by  the  fraction  |,  or  conversely  |. 

The  equality  of  two  ratios  is /Called  a  Proportion  ;  and 
the  terms  are  called  proportionals.  Thus,  2:4  =  3:6 
express  a  proportion,  signifying,  that  the  ratio  of  2  to  4 
is  equal  to  the  ratio  of  3  to  6. 

In  a  proportion,  the  first  and  fourth  terms,  that  is,  the 
antecedent  of  the  fii*st  ratio  and  the  consequent  of  the 
second,  are  called  the  extreme  terms  ;  and  the  second  and 
third  terms,  that  is,  the  consequent  of  the  first  ratio  and 
the  antecedent  of  the  second,  are  called  the  mean  terms. 
Thus,  in  the  proportion  3  :  9  =  4  :  12,  3  and  12  are 
the  extreme  terms,  9  and  4  the  mean  terms. 

It  is  to  be  observed  that,  if  four  numbers  be  in  propor- 
tion,  the  product  of  the  extreme  ter^ns  is  equal  to  the 
product  of  the  mean  terms. 


X.  proportiOjV.  205 

Since  the  product  of  the  extremes  in  every  proportion 
is  equal  to  the  product  of  the  means,  one  product  may 
be  taken  for  the  other.  Now,  if  we  divide  the  product  of 
the  extremes  by  one  extreme,  the  quotient  is  the  other 
extreme ;  therefore,  if  we  divide  the  product  of  the  means 
by  one  extreme,  the  quotient  is  the  other  extreme. 

To  apply  these  principles  to  practice,  let  it  be  asked — 
If  64  yards  of  cloth  cost  304  dollars,  what  will  36  yards 
cost  ?  In  the  first  place,  the  ratio  of  the  two  pieces  of 
*?loth  is  64  :  36 ;  and  secondly,  the  prices  are  in  the  same 
ratio  ;  that  is,  304  dollars  must  have  the  same  ratio  to  the 
price  of  36  yards,  that .64  yards  have  to  36  yards.  Now, 
if  we  put  A.  instead  of  the  answer,  we  shall  have  the  fol- 
lowing proportion,  64  ;  36  =3  304  :  A.  Here,  the  product 
of  the  means  is  10944,  which,  divided  by  64,  one  of  the 
extremes,  gives  the  quotient  171,  the  other  extreme,  which 
was  the  term  sought,  and  the  answer. 

Of  the  four  numbers  in  a  proportion,  two  are  of  one 
kind,  and  two  of  another.  In  the  preceding  example,  two 
of  the  terms  are  yards,  and  two  are  dollars. 

From  the  principles  of  ratio  and  proportion,  we  deduce 
The  Rule  of  Three  —  an  ancient  rule,  by  the  opera- 
tion of  which,  having  three  numbers  given,  we  find  a 
fourth,  which  has  the  same  'ratio  to  the  third  that  the 
second  has  to  the  first. 

RULE  OF  THREE.  MaJce  the  number,  tuJiich  is  of  the 
same  kind  ivith  the  answer,  the  third  term.  And  if,  from, 
the  nature  of  the  question,  the  fourth  term  or  answer  must 
be  greater  than  the  third  term,  maJce  the  greater  of  the  tioo 
remaining  terms  the  second  term,  and.  the  smaller  the  first; 
but,  if  the  fourth  term  must  he  less  than  the  third,  make 
the  less  of  the  two  remaiMng  terms  the  second,  term,  and. 
the  greater  the  first.  Multiply  the  second,  and  third  terms 
together,  and  divide  the  product  by  the  first  teim:  the 
quotient  will  be  the  fourth  term.,  or  answer. 

If  there  are  different  denominations  In  the  first  two  terms, 
they  must  both  be  reduced  to  the  lowest  denomination  in 
either  of  them ;  and  the  third  term  must  be  reduced  to  the 
lowest  denomination  mentioned  in  it. 


206  SUPPLEMENT.  X 

Operations  corresponding  to  the  Rule  of  Three  have 
already  been  taught,  in  Relations  of  Numbers,  Chap.  VI. 
To  show  the  correspondence,  suppose  it  to  be  asked — If  3 
yards  of  cloth  cost  4  dollars,  what  will  9  yards  cost  ? 


In  Relations  of  Numbers, 
the  question  stands  thus  — 
What  is  9  times  -J-  of  4? 


3)^ 

1 

9 


H 


12    Ans, 


In  the  Rule  of  Three,  the 
question  stands  thus — 
3:9  —  4:  what  number  ? 
3  :9=:4  :  A 
_9 

3)36" 

12    Ans. 


1.  If  I  buy  871  yards  of  cotton  cloth  for  78  dollars  39 

cents,  what  is  the  price  of  29  yards  of  the  same  ? 

371  :  29^:78.39  :  A  The  statements  of  this 

29  '  question  may  be  read  thus 

7  0551  — The  ratio  of  871  to  29 

15573  is  equal    to   the  ratio   of 

QT  1  \oo-yo  oi  /o  ^  1   .f       78.39  to  the  answer.     Or 

871  )2273.3 1(2.6 1^47/5.    .  .    Q-y,     i  •    .    on 

^  TjACf       ^  thus — As  871  yd.  is  to  29 

LLijL,  yd.,  so  is  $78.39  to  the 

^313  answer.      The   operation 

5226  amounts  to  nothing  more 

871  than  the  multiplication  of 

871  78.39  by  eVr 

.  2.  If  1-|  yard  of  cotton  cloth  cost  42  cents,  what  will 
87^  yards  cost,  at  the  same  price  per  yard  ? 
1.75  :  87.5  :=:  .42  :  A 

3.  If  I  can  buy  li  yard  of  cotton  cloth  for  6\  pence, 
how  many  yards  can  I  buy  for  £10  6s.  8d.  ? 

6d.  Iqr.  :  ^10  6s.  8d.=  1yd.  Iqr.  :  A 

4.  If  I  buy  54  barrels  of  flour  for  297  dollars,  what 
must  I  give  for  73  barrels,  at  the  same  rate  ? 

5.  If  7  workmen  can  do  a  piece  of  work  in  12  days, 
how  many  can  do  the  same  work  in  3  days  ? 

6.  If  20  horses  eat  70  bushels  of  oats  in  3  weeks,  how 
many  bushels  will  6  horses  eat  in  the  same  time  ? 

7.  If  a  piece  of  cloth  containing  76  yards  cost  136  dol- 
lars 80  cents,  what  is  that  per  ell  English  ? 


X.  PROPORTION.  207 

8.  If  a  staff  4  feet  long  cast  a  shadow  7  feet  in  length, 
on  level  ground,  what  is  the  height  of  a  steeple,  whose 
shadow  at  the  same  time  measures  198  feet  ? 

9.  How  many  yards  of  paper,  2^  feet  wide,  will  hang  a 
room,  that  is  20  yards  in  circuit,  and  9  feet  high  ? 

10.  A  certain  work  having  been  accomplished  in  12 
days,  by  working  4  hours  a  day,  in  what  time  might  it 
have  been  done  by  working  6  hours  a  day  ? 

11.  If  12  gallons  of  wine  are  worth  30  dollars,  what  is 
the  value  of  a  cask  of  wine,  containing  31^  gallons? 

12.  If  8 1  yards  of  cloth  cost  4  dollars  20  cents,  what 
will  13^  yards  cost,  at  the  same  rate? 

13.  How  many  yards  of  cloth  ^  yard  wide,  are  equal  to 
30  yards  1^  yard  wide  ? 

14.  If  7  pounds  of  sugar  cost  75  cents,  how  many 
pounds  can  I  buy  for  6  dollars  ? 

15.  If  2  pounds  of  sugar  cost  25  cents,  and  8  pounds 
of  sugar  are  worth  5  pounds  of  coffee,  what  will  100 
pounds  of  coffee  cost  ? 

16.  A  merchant  owning  f  of  a  vessel,  sold  f  of  his 
share,  {j  X  ^,)  for  957  dollars.  What  was  the  vessel 
worth,  at  that  rate  ? 

17.  A  merchant  failing  in  trade,  owes  62936  dollars  39 
cents ;  but  his  property  amounts  to  only  38793  dollars  96 
cents,  which  his  creditors  agreed  to  accept,  and  discharge 
him.  How  much  does  the  creditor  receive,  to  whom  he 
owes  2778  dollars  63  cents  ? 

18.  Bought  3  tons  of  oil,  for  503  dollars  25  cents  ;  85 
gallons  of  which  having  leaked  out.  I  wish  to  know  at 
what  price  per  gallon  I  must  sell  the  residue,  that  I  may 
neither  gain  nor  lose  by  the  bargain. 

19.  If,  when  the  pried  of  wheat  is  6s.  3d.  a  bushel,  the 
penny  loaf  weighs  9  oz.,  what  ought  it  to  weigh,  when 
wheat  is  at  8s.  2|d.  a  bushel  ? 

20.  If  15  yards  of  cloth  |  yard  wide  cost  6  dollars  25 
cents,  what  will  40  yards,  being  yard  wide,  cost  ? 

21.  Borrowed  of  a  friend  250  dollars  for  7  months; 
and  then,  to  repay  him  for  his  kindness,  I  loaned  him  300^ 
dollars.  How  long  must  he  keep  the  300  dollars,  to 
oalance  the  previous  favor? 


^s 


SUPPLEMEIMT, 


XI 


22.  If  4^  cwt.  be  carried  36  miles  for  §  5^,  how  many 
pounds  can  be  sent  20  miles  for  the  same  money  ? 

23.  A  person  owning  |  of  a  coal  mine 
share  for  570  dollars.     What  is  the  whole  mine  worth  ? 

24.  If  the  discount  on  §  106,  for  a  year,  be  $6,  what  is 
the  discount  on  $477,  for  the  same  time  ? 


sells  I  of  his 


111 


XI 

MEASUREMENT 

OF  SURFACES,  SOLIDS  AND  CAPACrnF.S. 

It  has  already  been  taught,  that  surfaces  are  measured 
m  squares,  and,  that  solid  bodies  are  measured  in  cubes. 

A  SQUARE  is  a  figure,  that  has  four 
equal  sides,  and  four  equal  angles.  Its 
angles  are  called  right  angles  :  angles 
more  pointed  are  called  acute  angles ; 
and  those  .less  pointed,  obtuse  angles. 
To  find  the  area  of  a  square,  in  smaller 
squares — Multiply  one  side  into  itself, 

1.  How  many  square  feet  are  there  in  a  table  that  meas- 
ures 4  feet  on  every  side?     How  many  square  inches? 

A  PARALLELOGRAM  is  a  four-sided 
figure,  having  opposite  sides  equal,  and 
having  four  right  angles.  To  find  the 
area  of  a  parallelogram — Multiply  the 
length  into  the  breadth. 

2.  How  many  square  rods  in  a  garden  measuring  4  rods 
in  length,  and  3  in  breadth  ?     How  many  square  feet? 

A  TRIANGLE  is  a  figure,  that  has  three 
sides  and  three  angles.  A  triangle, 
which  has  one  right  angle,  is  called  a 

RIGHT-ANGLED  TRIANGLE.       To  find  (he 

area  of  a  right-angled  triangle — Multi- 
ply the  base  by  half  the  perpendicular, 

3.  How  many  square  rods  are  there 
triangular  field,  measuring  98  rods  on 


rods  on  the  perpendicular?     How  many  acres? 


right-angled 
the  base,  and  75 


XL 


MKASUREMEINJT. 


209 


A  CIRCLE  is  a  plane  surface,  bound- 
ed by  one  curve  line,  called  the  circum-     a 
ference.     The  diameter  being  known,   . 
to   find    the   circumference  —  Multiply  'r 
the   diameter   by   3.14159.     Then,   to   ^ 
find  the  are^  —  Multiply  half  the  cir- 
cumference by  half  the  diameter. 

4.  How  many  square  inches  are  there  in  the  head  of  a 
barrel,  the  diameter  of  which  measures  17  inches  ? 

A  CUBE  is  a  regular  solid  body,  hav- 
ing six  equal,  square  sides.  To  find 
its  contents  in  smaller  cubes — Multiply 
the  breadth  of  a  side  twice  into  itself 
Tlie  product  of  the  length,  breadth,  and 
thickness  is  the  contents  of  any  thing, 
whose  opposite  sides  are  equal. 

5.  How  many  cubic  inches  are  there  in  a  box  measur- 
ing 34  inches  in  length,  26  in  width,  and  18  in  depth  ? 

A  CYLINDER  is  a  rouud  body,  with 
equal,  circular  ends.     To  find  its  cubical  ^^ 
contents — Find   the  area  of  one  end, 
and  multiply  this  by  the  length, 

6.  How  many  cubic  inches  are  there  in  a  drum  meas- 
uring 16  inches  across  the  head,  and  18  inches  in  lengdi  ? 

PLASTERING  and  PAVING  are  charged  by  the  square 
yard.  Their  surface  is  first  found  in  square  feet,  and  then 
reduced  to  square  yards. 

7.  How  many  square  yards  of  plastering  in  the  ceiling 
and  four  sides  of  a  room,  that  is  15  feet  long,  12  feet  wide, 
and  10  feet  high;  deducting  two  doors,  7  by  4  feet  each, 
and  four  windows,  5  by  3^^^  ^^^^  ^^^^^  ^ 

8.  How  many  bricks  are  required  to  pave  a  cellar,  that 
is  48  feet  long  and  30  feet  wide  ;  allowing  each  brick  to 
be  8  inches  long,  and  3.8  inches  wide  ?  Here  find  the 
area  of  the  cellar  in  square  inches,  and  divide  it  by  the 
square  inches  in  the  area  of  a  brick. 

SHINGLES  AND  CLAPBOARDS  are  of  various  dimen- 
sions.    Tliercfore,  to  know  how  many  are  requisite  to  cover 


210  SUPPLEMENT.  XI. 

a  building,  we  find  the  number  of  square  inches  in  the  roof 
or  side  to  be  covered,  and  divide  this  number  by  the  number 
of  square  inches,  that  one  shingle  or  clapboard  will  rover. 

9.  If  shingles  4  inches  in  width  be  laid  so  that  6  inches 
of  their  length  is  exposed  to  the  weather,  how  many  are 
required  to  cover  a  roof  45  by  32  feet  ? 

10.  How  many  clapboards,  each  covering 46  by 4  inch- 
es, are  sufficient  for  the  side  of  a  house  45  by  22  feet  ? 

BOARDS  are  sold  by  the  thousand  square  feet,  and 
each  board  is  measured  thus — Multiply  the  length  in  feet 
by  the  ividth  in  inches,  and  divide  the  product  hy  12; 
the  quotient  will  be  square  feet. 

11.  How  many  square  feet  are  there  in  17  boards,  each 
board  being  21  feet  long,  and  18.5  inches  wide  ? 

12.  How  many  square  feet  of  boards  will  floor  a  room 
14  by  18  feet,  allowing  -^^  of  the  stuff  for  waste? 

PLANK  AND  JOIST  are  measured  by  finding  how  many 
square  feet  of  boards,  one  inch  in  thickness,  they  are  equal 
to.  Therefore — Multiply  the  length  in  feet  by  the  width 
in  inches,  and  this  product  by  the  depth  in  inches;  then 
divide  the  last  product  by  12,  for  the  square  feet, 

13.  How  many  square  feet  in  a  plank  that  is  9  feet  in 
length,  14  inches  in  width,  and  2.4  inches  in  depth? 

14.  How  many  square  feet  in  a  joist  that  is  13  feet 
long,  4  inches  wide,  and  8.2  inches  deep  ? 

TIMBER  is  sold  by  the  cubic  ton.  To  measure  hewn 
timber — Multiply  the  length  in  feet  by  the  width  in 
inches,  and  this  product  hy  the  depth  in  inches ;  divide  by 
144,  for  the  cubic  feet,  and  then  by  50  for  the  tons. 

To  measure  round  timber — Take  the  circumference  in 
inches,  by  girding  the  log,  one-third  of  the  way  from  the 
but  to  the  top ;  then  multiply  the  length  in  feet,  by  the 
square  of  \  of  the  circumference ;  divide  by  144 /or  the 
cubic  feet,  and  then  by  40  for  the  tons, 

15.  How  much  hewn  timber  in  a  stick  measuring  25 
feet  in  length,  19  inches  in  width,  and  20  inches  in  depth  ? 

16.  How  much  round  timber  in  a  log,  30  feet  long,  anr 
55  inches  in  circumference  ? 


XI.  MEASUREMENT.  o|i 

CELLARS,  WELLS,  and  other  pits,  are  measured  by 
the  cube  of  six-feet  side ;  and  this  cube  is  called  a  square 
OF  EARTH.  To  measure  a  cellar — Add  together  the  depth 
of  the  four  corners^  divide  the  sum  by  4,  multiply  the  quo- 
tient by  the  length,  and  this  product  by  the  ividth,  all  in 
feet^  for  the  cubic  feet ;  then  divide  by  216  for  the  squares. 

To  measure  a  well — Proceed  as  ivith  a  cylinder  to  find 
the  cubic  feet,  and  divide  by  216  for  the  squares, 

17.  How  many  squares  in  a  cellar,  the  length  being  30 
ft.,  width  22  ft.,  depth  at  corners,  12  ft.,  9  ft.,  7  ft.,  and  4  ft.? 

18.  At  $1.08  a  square,  what  is  to  be  paid  for  digging 
a  well,  60  feet  deep,  and  8  feet  in  diameter  ? 

STONE  WALLS  are  measured  by  the  percA,  of  24|  cu- 
bic feet.  To  measure  a  straight  wall — Multiply,  in  feet, 
the  length  by  the  height,  and  this  product  by  the  thickness, 
for  the  cubic  feet ;  then  divide  by  24,1 5  for  the  perches. 

To  measure  a  circular  wall — Take  the  diameter,  to  the 
centre  of  the  thickness  of  the  wall,  and  compute  the  cir- 
cumference in  feet.  Then  multiply  the  circumference, 
height,  and  thickness  together,  all  in  feet,  for  the  cubic 
feet,  and  divide  by  24.75, /or  the  perches, 

19.  How  much  wall,  of  2  feet  thickness,  and  8  feet  height, 
in  a  cellar  measuring  36  feet  on  every  side  within  the  clear  r 

20.  How  much  wall  in  a  well  40  feet  deep  ;  the  wall 
being  2  feet  thick,  and  the  diameter  being  4.5  feet? 

BINS,  BOXES,  &:c.,  holding  commodities  sold  by  the 
gallon  or  bushel,  are  measured  thus — Find  the  contents  in 
cubic  inches,  as  already  taught ;  then  divide  by  231  fo% 
wine  gallons,  or,  by  2150.4 /or  bushels. 

21.  How  many  gallons  in  a  vat,  measuring  60  inches  in 
length,  .36  inches  in  breadth,  and  72  inches  in  depth  ? 

22.  How  many  bushels  of  grain  in  a  bin,  84  inches  in 
length,  32  inches  in  breadth,  and  48  inches  in  depth  ? 

CYLINDRIC  VESSELS,  such  as  tubs  and  cisterns  for 
holding  watei,  are  measured  thus — Multiply,  m  inches,  the 
diameter  of  one  end  into  itself,  and  this  product  into  the 
height ;  then  divide  by  294  for  the  wine  gallons. 


212  SUPPLEME.XT.  XIl. 

If  the  ends  of  the  vessel  be  unequal — Multiply  the 
greater  diameter  by  the  less,  and  to  the  product  add  i  of 
the  square  of  their  difference ;  rnidtiply  this  sum  by  the 
height,  and  divide  by  §94,  for  the  gallons, 

23.  How  many  gallons  will  a  tub  hold,  the  diameter  of 
which  is  18  inches,  and  the  height  22  inches  ? 

24.  How  many  gallons  of  water  will  a  cistern  hold, 
measuring  72  inches  across  the  bottom,  60  inches  across 
the  top,  and  84  inches  in  height  ? 

THE  CAPACITY  OF  CASKS  is  found  as  follows  — 
Take  the  interior  dimensions  as  nearly  as  possible,  Sub^ 
tract  the  diameter  of  the  head  from  the  diameter  at  the 
bung.  Multiply  the  difference  by  .7,  if  the  staves  be  much 
curved ;  or  by  .6,  if  little  curved ;  or  by  .65,  if  they  be 
o/ MEDIUM  curve.  Add  the  product  to  the  head,  diameter, 
and  the  sum  will  be  the  mean  diameter.  Square  the  mean 
diameter ;  multiply  the  square  by  the  length  of  the  cask, 
and  divide  this  product  by  294,  for  wine  gallons. 

25.  Find  the  number  of  gallons  in  a  cask  of  medium 
curve,  47  inches  in  length,  *3\  inches  diameter  at  the  bung, 
and  26  inches  diameter  at  the  head. 

26.  What  is  the  capacity  of  a  cask,  much  curved,  meas- 
uring 32.5  inches  in  length,  19  inches  at  the  bung,  and 
15.4  inches  at  the  head  ? 


xn. 

DUODECIMALS. 

Duodecimals  are  compound  numbers,  the  value  of 
whose  denominations  diminishes  in  a  uniform  ratio  of  12. 
They  are  applied  to  square  and  cubic  measure. 

The  denominations  of  duodecimals  are  the  foot,  (f), 
the  prime  or  inch,  Q,  the  second,  (''),  the  third,  ('''),  the 
fourth,  (''''),  the  fifth,  (^^^^0?  ^^^  ^^  ^"*  Accordingly,  the 
expression,  3  V  T*  9"'  ^^"'  denotes  3  feet  1  prime  7  sec- 
onds 9  thirds  6  fourths. 

The  accents,  used  to  distinguish  the  denominations  be* 
low  feet,  are  called  indices. 


Xn.  DUODECIMALS.  21? 

The  foot  being  viewed  as  the  unit,  duodecimals  present 
the  following  relations. 
1'     =  -J^  of  1  foot. 
V'    =z  ^^  of  -ji^  of  1  foot =1  -^   of  Koot 

1'^'    =  ^L  of  -5-I5  of  -^2  ^f    1   f^O^-       •       •   =  TT2Q    ^f   1   foot* 
1-'    :=  ^^  of  J^  of  T-V  of  j\  of   1  foot.    -20T^of  1  foOt. 

&c. 
Addition  and  subtraction  of  duodecimals  are  performed 
as  addition  and  subtraction  of  other  compound  numbers  ; 
12  of  a  lower  denomination  making  one  of  a  higher. 
Multiplication,  however,  when  both  the  factors  are  duo- 
decimals, is  peculiar,  and  will  now  be  considered. 

When  feet  are  multiplied  by  feet,  the  product  is  in  feet. 
For  instance,  if  required  to  ascertain  the  superficial  feet 
.  in  a  board  6  feet  long  and  2  feet  wide,  we  multiply  the 
length  by  the  breadth,  and  thus   find  its  superficial,  or 
square  feet  to  be  12.     But  when  feet  are  multiplied  by 
any  number  of  inches,  [primes],  the  effect  is  the  same  as 
that  of  multiplying  by  so  many  twelfths  of  a  foot,  and 
therefore  the  product  is  in  twelfths  of  a  foot,  or  inches. 
Thus  a  board,  6  feet  long  and  6  inches  wide,  contains  36 
mches,  because  the  length  being  multiplied  by  the  breadth, 
that  is,  6  feet  by  -^^  of  a  foot,  the  product  is  ||  of  a  foot, 
or  36^=3  feet.     When  feet  are  multiplied  by  seconds, 
the  product   is  in   seconds.     Thus  6  feet  multiplied   by 
6  seconds,  that  is,  -|  of  a  foot  by  -^^  of  tV  of  a  foot,  the 
product  is  -^  of  a  foot,  or  36^'=  3  inches. 
Feet  multiplied  by  feet,  produce  feet. 
Feet  multiplied  by  primes,  produce  primes. 
Feet  multiplied  by  seconds,  produce  seconds. 
Feet  multiplied  by  thirds,  produce  thirds. 

&c. 
Primes  multiplied  by  primes,  produce  seconds. 
Primes  multiplied  by  seconds,  produce  thirds. 
Primes  multiplied  by  thirds,  produce  fourths. 

&c. 
Seconds  multiplied  by  seconds,  produce  fourths. 
Seconds  multiplied  by  thirds,  produce  fifths. 
Seconds  multiplied  by  fourths,  produce  sixths 
he. 


6f. 

4'    8" 

4 

6'    5" 

2' 

7" 

11"/  4"" 

3 

2' 

4" 

0"' 

25 

6' 

8" 

214  SUPPLEMENT.  XII. 

If  we  would  find  the  square  feet  in  a  floor  6  f.  4'  8^^  in 
length,  and  4  f.  6'  h"  in  breadth,  we  proceed  as  follows. 

We  begin  on  the  right  hand, 
and  multiply  the  whole  multi- 
plicand, first  by  the  seconds  in 
the  multiplier,  then  by  the  inches, 
and  lastly  by  the  feet.  We  then 
add  the  results  together,  and  thus 
28f.ll'    7"    11-4""     obtain  the  answer. 

We  are  now  led  to  a  general  rule  for  the  multiplication 
of  duodecimal  numbers. 

RULE.  P/flfce  the  several  terms  of  the  multiplier  under 
the  corresponding  ones  of  the  m;ultiplicand.  Beginning 
on  the  right  hand,  multiply  the  several  terms  of  the  mul- 
tiplicand by  the  several  terms  of  the  multiplier  successive- 
ly, placing  the  right  hand  term  of  each  of  the  partial 
products  under  its  multiplier.  Then  add  the  partial  prod- 
ucts together;  observing  to  carry  one  for  every  twelve, 
both  in  multiplying  and  adding.  The  sum  of  the  partial 
products  ivill  be  the  answer. 

Questions  in  duodecimals  are  very  commonly  performed 
by  commencing  the  multiplication  with  the  highest  denom- 
ination of  the  multiplier,  jmd  placing  the  partial  products 
as  in  the  first  of  the  two  following  operations.  The  result 
is  the  same,  whichever  method  is  adopted.  The  second 
operation,  however,  is  according  to  the  rule  we  have  given, 
and  is  more  conformable  to  the  multiplication  of  numbers 
accompanied  by  decimals. 


3f. 

2' 

1" 

2f. 

6' 

4" 

6 

5' 

2" 

I 

7' 

3" 

&" 

1' 

0" 

10'"  4"" 

3f. 

2' 

1" 

2f. 

6' 

4" 

1' 

0" 

W" 

4"// 

1 

r 

3" 

6"' 

6 

5' 

2" 

8f.  I'  6''    A''' A"''  8f.   1'  6'^    A"'  A'"' 

When  there  are  not  feet  in  both  the  factors,  there  may 
not  be  any  feet  in  the  product ;  but,  after  what  has  been 
said,  there  will  be  no  difficulty  in  determining  the  places 
of  the  product. 


XII.  MISCELLANKOUS    EXAMPLES.  215 

1.  Multiply  14  r.  9'  by  4f.  €'. 

2.  What  are  the  contents  of  a  marble  slao,  whose  length 
IS  5f.  V,  and  breadth  If.  10'? 

3.  How  many  square  feet  are  there  in  the  floor  of  a  hall, 
48 f.  6'  long,  and  24 f.  3'  wide? 

4.  Multiply  4f.  T  8^'  by  9f.  6^ 

5.  How  many  square  feet  are  there  in  a  house  lot,  43  f 
3'  in  length,  and  25 f.  6'  in  breadth? 

6.  What  is  the  product  of  10  f.  4'  5''  by  7f.  8'  6"? 

7.  Calculate  the  square  feet  in  an  alley  44  f.  2'  9^'  long, 
and  2f.  10' 3'^  2''M'''' wide. 

8.  How  many  square  feet  are  there  in  a  garden,  39  f.  10' 
7"  long,  and  18  f.  8'  4"  wide  ? 

9.  What  is  the  product  of  24  f.  10'  8"  7'"  5""  by  9f. 
4'  6"  ? 

10.  Compute  the  solid  feet  in  a  wall,  53  f.  6'  long,  12  f. 
3' high,  and  2f.  thick. 

11.  The  length  of  a  ixx)m  is  20  feet,  its  breadth  14  feet 
6',  and  its  height  10  f.  4',  How  many  yards  of  painting 
are  there  in  its  walls,  deducting  a  fire  place  of  4f.  by  4f. 
4';  and  two  windows,  each  6i\  by  3f,  2'? 

12.  How  many  yards  of  carpeting,  yard  wide,  will  be 
required  for  a  room  21  f.  6'  long,  and  18 f.  wide? 

13.  What  will  the  plastering  of  a  ceiling  come  to,  at  10 
cents  a  square  yard,  supposing  the  length  21  feet  8  inches, 
and  the  breadth  14  feet  10  inches? 

14.  How  many  yards  of  papering  on  the  four  walls  of 
a  hall,  58  f.  8'  long,  21  f.  4'  wide,  and  13  f.  9'  high  ;  de- 
ducting  2  dooi*s,  each  7  f.  6'  high  and  4  f.  wide ;  7  windows, 
each  6f.  2'  high  and  3f.  10'  wide;  and  a  mop-board,  9 
inches  wide  around  the  hall  ? 


END    OF    PART    SECOND. 


/ 


216 


INDEX. 


ORAL  ARITHMETIC. 

Numeration   , 5 

Addition 9 

Subtraction     , iQ 

Correspondent  Examples  in  Addition  and  Subtraction 19 

Multiplication    24 

Division 31 

Correspondent  Examples  in  Multiplication  and  Division 37 

Connected  Operations  38 

Fractions    44 

Relations  of  Numbers   , 48 

Fractions  and  Relations    58 

Notation  of  Fractions 74 

WRITTEN  ARITHMETIC. 

Numeration   89 

Addition 93 

Subtraction 97 

Multiplication    103 

Abbreviations  in  Multiplication 105 

Division 108 

Abbreviations  in  Division ;  116 

Federal  Money   124 

Compound  Numbers    130 

Fractions    147 

Relations  of  Numbers 149 

Fractions  and  Relations     155 

Change  of  Whole  Numbers  to  Fractions , 156 

Change  of  Fractions  to  Whole  Numbers 157 

Percentage   ' 163 

Interest 165 

Discount 168 

Change  of  the  Terms  of  Fractions 169 

Fteduction  of  Fractions  to  Lower  Terms    169 

Compound  Fractions   170 

'  SUPPLEMENT. 


Cancellation  of  Factors 182 

Decimal  Fractions 183 

Change  of  Common  Fractions 

to  Decimals 188 

Ch.  of  Comp.  Num.  to  Dec. . .  188 
Ch.  of  Dec.  to  Comp.  Num. . .  188 
Exchange  of  Currencies  ....  189 

Percentage 101 

Commission 193 

Stocks 193 

Insurance 193 


Interest , . , 194 

Compound  Interest   198 

Discount 199 

Banking 19;) 

Profit  and  Loss 20C 

Partnership 201 

Assessment  of  Taxes 203 

Ratio  and  Proportion 20:) 

Rule  of  Three 205 

Measurement 208 

Duodecimals 21 2 


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EMERSON'S    SECOND   PART. 


Williams  College,  Oct.  2d,  1832. 
To  Mr.  Frederick  Emerson.  Sir,— 1  have  received  the  I \rsl 
artft  Second  Farts  o£ yoni  Nort^  American  Arithmetic;  and  ?m 
highly  pleased  v  ith  the  plan  of  the  work,  and  the  manner  of 
its  execution  ihv  ^  ^ar.  It  uniVes  simplicity  with  fulness,  and 
will  thus  be  sure  to  interest  the  beginner,  whilst  it  furnishes,  at 
the  same  time,  an  ample  guide  to  the  more  advanced  pupil. 
Respectfully  and  truly  yours,        ALBERT  HOPKINS. 

[Professor  of  Mathematies  and  Natural  Philosophy 
in  Williamstown  College.'] 

Union  College,  Schenectady,  Feb.  8th,  1833. 
Mr.  F.  Emerson.  Sir, — I  have  received  the  First  and  Second 
Parts  of  the  North  American  Arithmetic,  and  examined  them 
with  pleasure.  The  relations  of  numbers  are  happily  illusti^ated 
by  the  relations  of  sensible  objects.  This  renders  the  work 
more  entertaining  and  instructive,  especially  at  that  age  to  which 
arithmetic  is  better  adapted  than  many  of  those  studies  which 
for  want  of  a  system  like  this  are  too  oflen  introduced.  Here 
the  student  will  acquire  not  merely  rules  to  guide  his  hand,  but 
principles  to  enlighten  his  understanding.  He  is  not  furnished 
with  a  mere  mill  for  grinding  numbers  into  a  certain  result 
under  cover.         Wishing  you  continued  success, 

I  am,  sir.  respectfully,  your  ob't  ser't,        B.  F.  JOSLIN. 
[Professor  of  Mathematics  and  Natural  Philosovhy 
in  Union  College.] 

Burlington,  15th  February,  1833. 
{Condtision  of  a  l^'ter  to  the  Author.]  I  should  think  it  hardly 
possible  that  a  child  could  be  faithfully  conducted  through  these 
two  works,  [First  and  Second  Parts],  withoiit  being  vastly  better 
acquainted  with  the  subject  than  children  fo^.aerly  were.  Being 
judiciously  compelled  in  some  measure  to  invent  1  heir  own  rules, 
they  can  scarcely  fail  of  being  able  to  assign  a  proper  reason  for 
the  process,  as  well  as  to  recollect  it  for  future  use.  Indeed,  i 
do  not  know  any  one  particular  in  which,  for  the  use  of  very 
young  pupils,  they  could  be  improved.     Yours  respectfully, 

JAMES  DEAN 
[Professor  of  Mathematics  07id  Natural  Philosophy , 
in  the  University  of  Vermont.] 

Athens,  Oh^-  If'ebruary  the  1st,  1833. 
Mr.  Emerson.  Dear  Sir, — I  havb  carefully  and  attentively 
examined  the  •'  First  and  Second  Parts"  of  The  North  Amer- 
ican Arithmetic;"  and  can  say,  with  pleasure,  that  the  work 
meets  my  cordial  approbation.  You  have,  perhaps,  re  "  fered 
Arithmetic  as  highly  interesting  to  the  youthful  mind  aa  the 
nature  of  the  subject  will  admit.  Your  method  of  illustrating 
the  fundamental  priticiples  of  Fractions  is  clear,  forcible,  and 
peculiarly  happy  in  its  adaptation  to  the  minds  of  youth. 

1  am,  yours,  &c.        WlLLlAxM  WALL. 
[Professor  of  Mathematics ,  in  the  Ohio  University.] 

55_ 5S 


